the pessimistic diagnosabilities of some general regular graphs the pessimistic diagnosis strategy is a classic strategy based on the pmc model a system is t t diagnosable if provided the number of faulty processors is bounded by t all faulty processors can be isolated within a set of size at most t with at most one fault free node mistaken as a faulty one the pessimistic diagnosability of a system g denoted by t p g is the maximal number of faulty processors so that the system g is t t diagnosable in this paper we study the pessimistic diagnosabilities of some general k regular k connected graphs g n the main result t p g n 2 k 2 g under some conditions is obtained where g is the maximum number of common neighbors between any two adjacent vertices in g n as applications of the main result every pessimistic diagnosability of many famous networks including some known results such as the alternating group networks an n the k ary n cubes q n k the star graphs s n the matching composition networks g g 1 g 2 m and the alternating group graphs ag n are obtained we study the pessimistic diagnosabilities of some general regular graphs under the pmc model the exact value of t p g n is obtained as applications the pessimistic diagnosability of an n q n k s n some mcn and ag n etc are obtained

equal relation between the extra connectivity and pessimistic diagnosability for some regular graphs abstract extra connectivity and the pessimistic diagnosis are two crucial subjects for a multiprocessor system s ability to tolerate and diagnose faulty processor the pessimistic diagnosis strategy is a classic strategy based on the pmc model in which isolates all faulty vertices within a set containing at most one fault free vertex in this paper the result that the pessimistic diagnosability t p g equals the extra connectivity 1 g of a regular graph g under some conditions are shown furthermore the following new results are gotten the pessimistic diagnosability t p s n 2 4 n 9 for split star networks s n 2 t p n 2 n 4 for cayley graphs generated by transposition trees n t p n 4 n 11 for cayley graph generated by the 2 tree n t p b p n 2 n 2 for the burnt pancake networks b p n as corollaries the known results about the extra connectivity and the pessimistic diagnosability of many famous networks including the alternating group graphs the alternating group networks bc networks the k ary n cube networks etc are obtained directly

on maximal 3 restricted edge connectivity and reliability analysis of hypercube networks abstract it is shown in this work that all n dimensional hypercube networks for n 4 are maximally 3 restricted edge connected employing this observation we analyze the reliability of hypercube networks and determine the first 3 n 5 coefficients of the reliability polynomial of n cube networks

on 3 extra connectivity and 3 extra edge connectivity of folded hypercubes given a graph mbi g and a non negative integer g the g extra connectivity resp g extra edge connectivity of mbi g is the minimum cardinality of a set of vertices resp edges in mbi g if it exists whose deletion disconnects mbi g and leaves each remaining component with more than g vertices this study shows that the 3 extra connectivity resp 3 extra edge connectivity of an mbi n dimensional folded hypercube is 4 n 5 for n geq 6 resp 4 n 4 for n geq 5 this study also provides an upper bound for the g extra connectivity on folded hypercubes for g geq 6

recursive circulants and their embeddings among hypercubes we propose an interconnection structure for multicomputer networks called recursive circulant recursive circulant g n d is defined to be a circulant graph with n nodes and jumps of powers of d g n d is node symmetric and has some strong hamiltonian properties g n d has a recursive structure when n cdm 1 c d we develop a shortest path routing algorithm in g cdm d and analyze various network metrics of g cdm d such as connectivity diameter mean internode distance and visit ratio g 2m 4 whose degree is m compares favorably to the hypercube qm g 2m 4 has the maximum possible connectivity and its diameter is 3m 1 4 recursive circulants have interesting relationship with hypercubes in terms of embedding we present expansion one embeddings among recursive circulants and hypercubes and analyze the costs associated with each embedding the earlier version of this paper appeared in park and chwa proc internat symp parallel architectures algorithms and networks ispan 94 kanazawa japan december 1994 pp 73 80

structure connectivity and substructure connectivity of k ary n cube networks abstract the k ary n cube is one of the most attractive interconnection networks for parallel and distributed computing system in this paper we investigate the fault tolerant capabilities of k ary n cubes with respect to the structure connectivity and substructure connectivity let h is a connected graph the structure connectivity of a graph g denoted by g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to h the substructure connectivity of a graph g denoted by s g h is the minimum cardinality of a set of connected subgraphs in g whose deletion disconnects the graph g and every element in the set is isomorphic to a connected subgraph of h we show q n k h and s q n k h for each h k1 k1 1 k1 2 k1 3

a new proof for exact relationship between extra connectivity and extra diagnosability of regular connected graphs under mm model abstract the extra connectivity and extra diagnosability are two important measures for network reliability under mm model two possible relationships between extra connectivity and extra diagnosability were proposed in reference 24 however there are some shortcomings in it 1 the conclusion of theorem 3 9 is wrong 2 the corresponding proof of theorem 3 9 is flawed 3 the exact relationship is still not clear in this paper we impose reasonable constraints and give a new and correct proof for measuring the exact relationship between h extra connectivity h g and h extra diagnosability t h m g of the regular connected graph g under mm model which is t h m g h g h as an application we directly obtain the h extra diagnosability of star graph s n alternating group graph network a n n and n k star graph s n k by our proposed new result

lee distance and topological properties of k ary n cubes in this paper we consider various topological properties of a k ary n cube q sub n sup k using lee distance we feel that lee distance is a natural metric for defining and studying a q sub n sup k after defining a q sub n sup k graph using lee distance we show how to find all disjoint paths between any two nodes given a sequence of radix k numbers a function mapping the sequence to a gray code sequence is presented and this function is used to generate a hamiltonian cycle embedding the graph of a mesh and the graph of a binary hypercube into the graph of a q sub n sup k is considered using a k ary gray code we show the embedding of a k n sub 1 spl times k n sub 2 spl times spl times k n sub m dimensional mesh into a q sub n sup k where n spl sigma sub i l sup m n sub i then using a single digit 4 ary reflective gray code we demonstrate embedding a q sub n into q sub n 2 sup 4 we look at how lee distance may be applied to the problem of resource placement in a q sub n sup k by using a lee distance error correcting code although the results in this paper are only preliminary lee distance error correcting codes have not been applied previously to this problem finally we consider how lee distance can be applied to message routing and single node broadcasting in a q sub n sup k in this section we present two single node broadcasting algorithms that are optimal when single port and multi port i o is used

a formula to compute the microcanonical volume of reactive initial conditions in transition state theory we present the formal proof of a procedure to compute the phase space volume of initial conditions for trajectories that for a constant energy escape or react from a multi dimensional potential well with one or several exit entrance channels the procedure relies on a phase space formulation of transition state theory it gives the volume of reactive initial conditions as the sum over the exit entrance channels where each channel contributes by the product of the phase space flux associated with the channel and the mean residence time in the well of those trajectories which escape through the channel an example is given to demonstrate the computational efficiency of the procedure

diagnosability for two families of composition networks abstract diagnosability is an important parameter to assess the reliability of multiprocessor systems a multiprocessor system is called t diagnosable if all faulty processors can be identified without replacement as long as the number of faulty processors does not exceed t the diagnosability is the maximum value of t such that multiprocessors system is t diagnosable matching composition networks mcns and cycle composition networks ccns are two typical network structures to construct multiprocessor systems let g be a mcn which is obtained by adding a perfect matching between two graph g 0 and g 1 and let h be a ccn obtained by adding a perfect matching between g i and g i 1 mod m for each i 0 1 m 1 with m 3 in this paper we show that the diagnosability of g under the pmc g 2 model and mm model g 5 is n 1 if g 0 g 1 k n otherwise g furthermore we show that the diagnosability of h under the pmc model h 4 and mm model h 5 is h

how many triangles and quadrilaterals are there in an n dimensional augmented cube abstract the augmented cube is an important variant of hypercube as interconnection topology of parallel computing in this paper we examine the numbers of short cycles in augmented cubes and prove that for n 3 there are 2 n n 1 triangles and 2 n 2 2 n 2 5 n 11 quadrilaterals in an n dimensional augmented cube this result shows that augmented cubes are promising interconnection networks with superior connectivity and fault tolerant capability

direct construction of a dividing surface of minimal flux for multi degree of freedom systems that cannot be recrossed the fundamental assumption of transition state theory is the existence of a dividing surface having the property that trajectories originating in reactants resp products must cross the surface only once and then proceed to products resp reactants recently it has been shown wiggins et al 2001 phys rev lett 86 5478 uzer et al 2002 nonlinearity 15 957 how to construct a dividing surface in phase space for hamiltonian systems with an arbitrary finite number of degrees of freedom having the property that trajectories only cross once locally in this letter we provide an argument showing that the flux across this dividing surface is a minimum with respect to certain types of variations of the dividing surface

asymptotically optimal algorithms for one to one pickup and delivery problems with applications to transportation systems pickup and delivery problems pdps in which objects or people have to be transported between specific locations are among the most common combinatorial problems in real world logistical operations a widely encountered type of pdp is the stacker crane problem scp where each commodity customer is associated with a pickup location and a delivery location and the objective is to find a minimum length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in immediate consecutive order the scp is np hard and the best known approximation algorithm only provides a 9 5 approximation ratio in this paper we examine an embedding of the scp within a stochastic framework and our objective is three fold first we describe a large class of algorithms for the scp where every member is asymptotically optimal i e it produces almost surely a solution approaching the optimal one as the number of pickups deliveries goes to infinity moreover one can achieve computational complexity o n2 e within the class where n is the number of pickup delivery pairs and e is an arbitrarily small positive constant second we characterize the length of the optimal scp tour asymptotically finally we study a dynamic version of the scp whereby pickup and delivery requests arrive according to a poisson process and which serves as a model for large scale demand responsive transport drt systems for such a dynamic counterpart of the scp we derive a necessary and sufficient condition for the existence of stable vehicle routing policies which depends only on the workspace geometry the distributions of pickup and delivery points the arrival rate of requests and the number of vehicles our results leverage a novel connection between the euclidean bipartite matching problem and the theory of random permutations and for the dynamic setting exhibit novel features that are absent in traditional spatially distributed queueing systems

connectivity of large wireless networks under a general connection model this paper studies networks where all nodes are distributed on a unit square a 1 2 1 2 2 following a poisson distribution with known density and a pair of nodes separated by an euclidean distance x are directly connected with probability gr x g x r independent of the event that any other pair of nodes are directly connected here g 0 0 1 satisfies the conditions of rotational invariance nonincreasing monotonicity integral boundedness and g x o 1 x2log2x further r log b c where c 2g x dx and b is a constant denote the aforementioned network by g x gr a we show that as 1 the distribution of the number of isolated nodes in g x gr a converges to a poisson distribution with mean e b 2 asymptotically almost surely a a s there is no component in g x gr a of fixed and finite order k 1 c a a s the number of components with an unbounded order is one therefore as the network a a s contains a unique unbounded component and isolated nodes only a sufficient and necessary condition for g x gr a to be a a s connected is that there is no isolated node in the network which occurs when b as these results expand recent results obtained for connectivity of random geometric graphs from the unit disk model and the fewer results from the log normal model to the more general and more practical random connection model

structure fault tolerance of hypercubes and folded hypercubes abstract let g be a graph and t be a certain connected subgraph of g the t structure connectivity g t resp t substructure connectivity s g t of g is the minimum number of a set of subgraphs f t 1 t 2 t m resp f t 1 t 2 t m such that t i is isomorphic to t resp t i is a connected subgraph of t for every 1 i m and f s removal will disconnect g let q n and f q n denote the n dimensional hypercube and folded hypercube respectively in 12 the q n t and s q n t were determined for t k 1 1 k 1 2 k 1 3 c 4 in this paper we generalize the above results by determining q n t and s q n t for t p k c 2 k k 1 4 where 3 k n we also determine f q n t and s f q n t for t p k c 2 k k 1 3 where n 7 and 2 k n

super connectivity of balanced hypercubes abstract the reliability of an interconnection network is an important issue for multiprocessor systems in this paper we study a reliability measure called super connectivity in the balanced hypercube bh n which is a variant of the hypercube we show that the super connectivity of bh n is 4 n 4 and the super edge connectivity of bh n is 4 n 2 for n 2 that is to become a disconnected graph containing no isolated vertex we need to remove at least 4 n 4 vertices resp 4 n 2 edges from bh n

on the geometry of transport in phase space i transport in k degree of freedom hamiltonian systems 2 k abstract over the past several years the effect of geometrical structures such as kam tori resonance zones and cantori on phase space transport in two degree of freedom hamiltonian systems has been studied in three degree of freedom systems and higher these structures no longer form complete or partial barriers to transport in the sense that they or their associated stable and unstable manifolds are no longer codimension one in the level set of the hamiltonian in this paper we show that in the 2 k 1 dimensional level sets of the hamiltonian of a class of k degree of freedom hamiltonian systems the 2 k 2 dimensional stable and unstable manifolds of normally hyperbolic invariant 2 k 3 dimensional spheres form partial barriers to transport under certain conditions we show that lobes and turnstiles can be formed from such geometrical structures in a way that is exactly analogous to the situation for two degree of freedom hamiltonian systems considering a foliation of the level set of the hamiltonian into disjoint regions separated by such partial barriers we can then give exact formulas for the transport of volumes of phase space throughout the different regions based on the dynamics of these generalized lobes and turnstiles on the other hand despite the fact that these partial barriers are codimension one for k degree of freedom systems k 3 they may not intersect in such a way so as to divide the phase space

2 3 restricted connectivity of locally twisted cubes given a graph g and non negative integer h the h restricted connectivity of g is the minimum cardinality of a set of nodes in g if exists whose deletion disconnects g and the degree of each node in every remaining component is at least h the h restricted connectivity is a generalization of the classical connectivity and can provide more accurate measures for the reliability or fault tolerance of multiprocessor system the n dimensional locally twisted cubes denoted by ltq n are a well known network topology for building multiprocessor systems in this paper we first show that 2 restricted connectivity of the n dimensional locally twisted cubes is 4 n 8 for n 4 and show that 3 restricted connectivity is equal to 8 n 24 for n 5

conditional connectivity of recursive interconnection networks respect to embedding restriction abstract large scale multiprocessor systems always take some recursive interconnection networks as underlying topologies let g n be an n dimensional recursive interconnection network the m embedding restricted connectivity m g n resp the m embedding restricted edge connectivity m g n of g n is the cardinality of a minimum subset of nodes resp edges if any whose deletion disconnects g n and each node of the remaining components lies in an undamaged m dimensional sub network g m in this paper we present some relationships between the proposed indices and other conditional connectivity indices in general recursive interconnection networks we give some bounds on these two indices in k ary n cubes and bubble sort networks in addition we determine these two indices in k ary n cubes and bubble sort networks in some cases

linearly many faults in cayley graphs generated by transposition trees abstract we prove that when linearly many vertices are deleted in a cayley graph generated by a transposition tree the resulting graph has a large connected component containing almost all remaining vertices

the extra connectivity of bubble sort star graphs connectivity plays an important role in measuring the fault tolerance of a multiprocessor system in the case of vertices failures extra connectivity is an important indicator of a network s ability for diagnosis and fault tolerance in this paper we analyze the fault tolerance ability of the bubble sort star network denoted by b s n a well known interconnection network proposed for multiprocessor systems and establish the g extra connectivity for 1 g 3

the largest component of faulty star graphs abstract the minimum size of the largest component of a network with faults is a useful parameter to make a full evaluation of this network and it is helpful for estimating some other parameters in graph theory in this paper we focus on discussing the minimum size of the largest component of star graph s n which contains faulty edges specifically we show that there exists at most one two three four five vertices beyond the largest component of s n f with f e s n and f 2 n 5 3 n 8 4 n 11 5 n 15 6 n 19 respectively as applications we study the g extra edge connectivity of s n for 1 g 5 and study the strong menger edge connectivity of s n f with f e s n

a two stages tree searching algorithm for finding three completely independent spanning trees abstract for the underlying graph g of a network k 2 spanning trees of g are called completely independent spanning trees cists for short if they are mutually inner node disjoint it is well known that determining the existence of k cists in a graph is an np hard problem even for k 2 also there exists a k connected graph which does not possess two cists for any k 2 accordingly researches focused on the problem of constructing multiple cists in some particular famous networks pai and chang 2016 proposed a unified approach to recursively construct two cists with diameter 2 n 1 in several n dimensional hypercube variant networks for n 4 including locally twisted cubes l t q n and crossed cubes c q n later on new constructions showed that the diameter of two cists can be reduced to 2 n 2 if n 4 and 2 n 3 if n 5 for l t q n and to 2 n 2 if n 4 5 and 2 n 3 if n 6 for c q n in this paper we intend to construct more cists for those networks that are paid attention we develop a novel tree searching algorithm called two stages tree searching algorithm abbreviated as ts2 which can be used to construct three cists of a 6 regular graph if the scale of the graph is not too large and there exist three cists for certain in particular we demonstrate that three cists of l t q 6 and c q 6 can be acquired by using ts2 moreover for high dimensional l t q n and c q n with n 7 we show that their three cists can be constructed by recursion consequently we obtain the following results i for l t q n the diameters of three cists we constructed are 11 12 and 12 when n 6 and all are 2 n 1 when n 7 ii for c q n all diameters of three cists we constructed are 12 when n 6 and 2 n 1 when n 7

structure fault tolerance of balanced hypercubes abstract the s structure connectivity g s and s substructure connectivity s g s of graph g with structure graph s which are newly proposed in recent years can better measure fault tolerance of networks in this paper we study b h n s and s b h n s of n dimensional balanced hypercube b h n which is a relatively popular network topology proposed recently for s p k c 6 k 1 r where 4 k n 6 6 n and 4 r 5

geometry helps in matching a set of 2n points on the plane induces a complete weighted undirected graph as follows the points are the vertices of the graph and the weight of an edge between any two points is the distance between the points under some metric the problem of finding a minimum weight complete matching mwcm in such a graph is studied an o n 2 5 log n 4 algorithm is given for finding an mwcm in such a graph for the l 1 manhattan the l 2 euclidean and the l infty metrics the bipartite version of the problem is also studied where half the points are painted with one color and the other half with another color and the restriction is that a point of one color may be matched only to a point of another color an o n 2 5 log n algorithm for the bipartite version for the l 1 l 2 and l infty metrics is presented the running time for the bipartite version can be further improved to o n 2 log n 3 for the l 1 and l infty metrics

optimal role and position assignment in multi robot freely reachable formations abstract many multi robot problems require the achievement of formations as part of the overall mission this work considers a scenario in which unlabeled homogeneous robots must adopt a given formation pattern buildable anywhere in the environment this involves finding the relative pose of the formation in regard to the initial robot positions understood as a translation and a rotation and the optimal assignment of the role of each robot within the formation this paper provides an optimal solution for the combined parameters of translation rotation and assignment that minimizes total displacement to achieve this objective we first formally prove that the three decision variables are separable since computing the optimal assignment without accounting for the rotation is a computationally expensive problem we propose an algorithm that efficiently computes the optimal roles together with the rotation the algorithm is provably correct and finds the optimal solution in finite time a distributed implementation is also discussed simulation results characterize the complexity of our solution and demonstrate its effectiveness

fault tolerant strong menger edge connectivity and 3 extra edge connectivity of balanced hypercubes abstract a connected graph g is called strongly menger edge connected if for any two distinct vertices x y of g there are min deg g x deg g y edge disjoint paths between x and y the g g 0 extra edge connectivity of the connected graph g denoted by g g is the minimal cardinality of a set of edges of g if exists whose deletion disconnects g and each remaining component contains more than g vertices in this paper we show that the n dimensional balanced hypercube b h n which is a variant of hypercube q n is still strongly menger edge connected even when there are 2 n 4 faulty vertices resp 2 n 2 faulty edges for n 2 moreover under the restricted condition that each vertex has at least two fault free edges we show that b h n is still strongly menger edge connected even when there are 6 n 8 faulty edges for n 2 these results are all optimal with respect to the maximum number of tolerated edge faults furthermore we showed that the 3 extra edge connectivity of b h n is 8 n 8 for n 2

symmetric property and reliability of balanced hypercube huang and wu in ieee transactions on computers 46 1997 484 490 introduced the balanced hypercube bh n as an interconnection network topology for computing systems and they proved that bh n is vertex transitive however some other symmetric properties say edge transitivity and arc transitivity of bh n remained unknown in this paper we solve this problem and prove that bh n is an arc transitive cayley graph using this we also investigate some reliability measures including super connectivity cyclic connectivity etc in bh n first we prove that every minimum edge cut of bh n n ge 2 isolates a vertex and every minimum vertex cut of bh n n ge 3 isolates a vertex this is stronger than that obtained by wu and huang which shows the connectivity and edge connectivity of bh n are 2n second yang applied mathematics and computation 219 2012 970 975 proved that for n ge 2 the super connectivity of bh n is 4n 4 and the super edge connectivity of bh n is 4n 2 in this paper we proved that bh n n ge 2 is super lambda prime but not super kappa prime that is every minimum super edge cut of bh n n ge 2 isolates an edge but the minimum super vertex cut of bh n n ge 2 does not isolate an edge third we also obtain that for n ge 2 the cyclic connectivity of bh n is 4n 4 and the cyclic edge connectivity of bh n is 4 2n 2 that is to become a disconnected graph which has at least two components containing cycles we need to remove at least 4n 4 vertices resp 4 4n 2 edges from bh n n ge 2

the extra connectivity of the enhanced hypercubes abstract as one of the best parameters to evaluate the fault tolerance of interconnection networks extra connectivity plays an important role in multiprocessor systems for reliable computing given a graph g and a non negative integer g the g extra connectivity resp g extra edge connectivity of g denoted by g g resp g g is the minimum cardinality of a set of vertices resp edges in g if exists whose deletion disconnects g and leaves each remaining component with at least g 1 vertices in this paper we determine g q n k and g q n k of the enhanced hypercube q n k for g 1 2 furthermore as the byproduct of our results we determine the conditional diagnosability of q n k under the comparison model and pmc model

reliability measures in relation to the h extra edge connectivity of folded hypercubes the folded hypercube fq n as a variation of the hypercube q n was proposed by a el amawy and s latifi in 1991 the h extra edge connectivity of the underlying topological graph of a multiprocessor system is a kind of measure for the reliability of the multiprocessor system in this paper we determine the exact value of h fq n for integer h 1 h 2 n 2 1 and 6 n which generalizes several known results for h n more interestingly we also show that h fq n is the constant n 2 r 1 2 n 2 r for 2 n 2 r l r h 2 n 2 r where r 1 2 n 2 1 and l r 2 2 r 1 3 if n is odd and l r 2 2 r 1 2 3 if n is even in particular for r n 2 1 2 n 2 3 h 2 n 1 h fq n 2 n

structure connectivity and substructure connectivity of twisted hypercubes abstract let g be a graph and t a certain connected subgraph of g the t structure connectivity g t or resp t substructure connectivity s g t of g is the minimum number of a set of subgraphs f t 1 t 2 t m or resp f t 1 t 2 t m such that t i is isomorphic to t or resp t i is a connected subgraph of t for every 1 i m and f s removal will disconnect g the twisted hypercube h n is a new variant of hypercubes with asymptotically optimal diameter introduced by zhu in this paper we will determine both h n t and s h n t for t k 1 r p k respectively where 3 r 4 and 1 k n

monotonic target assignment for robotic networks consider an equal number of mobile robotic agents and distinct target locations dispersed in an environment each agent has a limited communication range and either 1 knowledge of every target position or 2 a finite range sensor capable of acquiring target positions and no a priori knowledge of target positions in this paper we study the following target assignment problem design a distributed algorithm with which the agents divide the targets among themselves and simultaneously move to their unique target we evaluate an algorithm s performance by characterizing its worst case asymptotic time to complete the target assignment that is the task completion time as the number of agents and targets increases and the size of the environment scales to accommodate them we introduce the intuitive class of monotonic algorithms and give a lower bound on its worst case completion time we design and analyze two algorithms within this class the etsp assgmt algorithm which works under assumption 1 and the grid assgmt algorithm which works under either assumption 1 or 2 in ldquosparse environments rdquo where communication is infrequent the etsp assgmt algorithm is within a constant factor of the optimal monotonic algorithm for worst case initial conditions in ldquodense environments rdquo where communication is more prevalent the grid assgmt algorithm is within a constant factor of the optimal monotonic algorithm for worst case initial conditions in addition we characterize the performance of the grid assgmt algorithm for uniformly distributed targets and agents and for the case when there are more agents than targets

phase synchronization control of complex networks of lagrangian systems on adaptive digraphs this paper presents a formation control and synchronization method that utilizes adaptive network topologies for a class of complex dynamical networks comprised of a large number of highly nonlinear euler lagrange el systems a time varying and switching network topology constructed by the adaptive graph laplacian matrix relaxes the standard requirement of consensus stability even permitting exponential synchronization on an unbalanced digraph or a weakly connected digraph that can sporadically lose connectivity the time varying graph laplacian matrix is adapted by an adaptive control scheme based on relative positions and errors of synchronization and tracking the adaptive graph laplacian is integrated with a phase synchronization controller that synchronizes the relative motions of el systems moving in elliptical orbits thereby yielding a smaller synchronization error than an uncoupled tracking control law in the presence of bounded disturbances and modeling errors an example of reconfiguring hundreds of spacecraft in low earth orbit shows the effectiveness of the proposed phase synchronization controller for a large number of complex el systems moving in periodic elliptical orbits

sufficient conditions for k restricted edge connected graphs for a connected graph g v e an edge set s e is a k restricted edge cut if g s is disconnected and every component of g s has at least k vertices the k restricted edge connectivity of g denoted by k g is defined as the cardinality of a minimum k restricted edge cut let k g min x x x k g x is connected where x v x g is maximally k restricted edge connected k optimal for short if k g k g the k restricted edge connectivity is more refined network reliability indices than edge connectivity in this paper let k 2 be an integer and let g be a graph of order g at least 2k satisfying n u n v 2 k 2 for all pairs u v of nonadjacent vertices if for each triangle t there exists at least one vertex v v t such that d v g 2 k 1 then g is k optimal

relating the extra connectivity and the conditional diagnosability of regular graphs under the comparison model extra connectivity and conditional diagnosability are two crucial subjects for a multiprocessor system s ability to tolerate and diagnose faulty processors the extra connectivity and the conditional diagnosability of many well known multiprocessor systems have been widely investigated in this paper the relationship between the extra connectivity and the conditional diagnosability of regular graphs is explored we establish that the conditional diagnosability under the comparison model is equal to the 2 extra connectivity finally we give empirical analysis on the extra connectivity and conditional diagnosability of some graphs by our proposed relationship we propose the fault tolerant properties of regular graphs with some basic conditions the conditional diagnosability under the comparison model is equal to the 2 extra connectivity the 2 extra connectivity of sn is 3 n 7 by our proposed relationship the 2 extra connectivity of cayley graphs generated by transposition trees is 3 n 8

strongly menger edge connectedness and strongly menger vertex connectedness of regular networks abstract let f be a conditional faulty edge set of a graph g here the condition imposed on f is g f r for some fixed integer r the graph g is called f strongly menger edge connected of order r if each pair of distinct vertices u and v are connected by min deg g f u deg g f v edge disjoint paths in g f where deg g f u and deg g f v are the degrees of u and v in g f respectively a graph g is t strongly menger edge connected of order r if g is f strongly menger edge connected of order r for every f e g with f t one can consider the vertex version in an analogous way there are a number of papers with results for r 2 for several popular classes of interconnection network in this paper we develop a unified approach by considering various sufficient conditions of a regular graph to be f strongly menger edge connected of order r and the corresponding vertex version as corollaries several results on fundamental classes of interconnection networks are presented including new results as well as strengthening of existing results such as the one given by qiao and yang 2017 20

restricted connectivity for three families of interconnection networks vertex connectivity and edge connectivity are two important parameters in interconnection networks even though they reflect the fault tolerance correctly they undervalue the resilience of large networks by the concept of conditional connectivity and super connectivity the concept of restricted vertex connectivity and restricted edge connectivity of graphs was proposed by esfahanian a h esfahanian generalized measures of fault tolerance with application to n cube networks ieee transactions on computers 38 1989 1586 1591 such measures take the resilience of large networks into consideration in this paper we propose three families of interconnection networks and discuss their restricted vertex connectivity and restricted edge connectivity in particular the hypercubes twisted cubes crossed cubes mobius cubes star graphs pancake graphs recursive circulant graphs and k ary n cubes are special cases of these families

the extra connectivity extra conditional diagnosability and t k diagnosability of the data center network dcell abstract connectivity and diagnosability are two important metrics in evaluating the fault tolerability of a network the g extra connectivity and the g extra conditional diagnosability are both defined under the restraint that every component of the network removing a faulty vertex set has at least g 1 fault free vertices the t k diagnosability is an outstanding diagnosis strategy in which the identified faulty vertex set is allowed to contain at most k fault free vertices as a well known model for a large scale data center network dcn with a server centric structure the m dimensional dcell with n port switches and t m n servers d m n has many desirable properties in this paper we first investigate the g extra connectivity of d m n for 0 g n 1 based on this we establish the g extra conditional diagnosability of d m n under the pmc model for 0 g n 1 finally we evaluate the t k diagnosability of d m n under the pmc model for 1 k n 1

relationship between conditional diagnosability and 2 extra connectivity of symmetric graphs the conditional diagnosability and the 2 extra connectivity are two important parameters to measure ability of diagnosing faulty processors and fault tolerance in a multiprocessor system the conditional diagnosability t c g of g is the maximum number t for which g is conditionally t diagnosable under the comparison model while the 2 extra connectivity 2 g of a graph g is the minimum number k for which there is a vertex cut f with f k such that every component of g f has at least 3 vertices a quite natural problem is what is the relationship between the maximum and the minimum problem this paper partially answers this problem by proving t c g 2 g for a regular graph g with some acceptable conditions as applications the conditional diagnosability and the 2 extra connectivity are determined for some well known classes of vertex transitive graphs including star graphs n k star graphs alternating group networks n k arrangement graphs alternating group graphs cayley graphs obtained from transposition generating trees bubble sort graphs k ary n cube networks dual cubes pancake graphs and hierarchical hypercubes as well furthermore many known results about these networks are obtained directly we reveal the relationship between conditional diagnosability and 2 extra connectivity of graphs the conditional diagnosability under the comparison model is equal to the 2 extra connectivity as applications these parameters are determined for some well known classes of graphs

on connectivity of the cartesian product of two graphs in this paper sufficient conditions for the cartesian product of two graphs to be maximum edge connected maximum point connected super edge connected or super point connected are presented as a corollary some graphs with these properties are obtained furthermore the relationship between super point connectivity and super edge connectivity on the cartesian product of graphs is pointed out

edge fault tolerance analysis of super k restricted connected networks an edge cut x of a connected graph g is a k restricted edge cut if g x is disconnected and every component of g x has at least k vertices additionally if the deletion of a minimum k restricted edge cut isolates a connected component of k vertices then the graph is said to be email protected k in this paper several sufficient conditions yielding email protected k graphs are given in terms of the girth and the diameter

the multi agent rendezvous problem part 2 the asynchronous case this paper is concerned with the collective behavior of a group of n 1 mobile autonomous agents labelled 1 through n which can all move in the plane each agent is able to continuously track the positions of all other agents currently within its sensing region where by an agent s sensing region we mean a closed disk of positive radius r centered at the agent s current position the multi agent rendezvous problem is to devise local control strategies one for each agent which without any active communication between agents cause all members of the group to eventually rendezvous at a single unspecified location this paper describes a family of unsynchronized strategies for solving the problem correctness is established appealing to the concept of analytic synchronization

linearly many faults in cayley graphs generated by transposition triangle free unicyclic graphs abstract we consider a kind of cayley graphs including the modified bubble sort graphs m b n the cayley graphs generated by lollipop graphs h n n 1 and the other cayley graphs generated by unicyclic triangle free graphs we get that if we delete linearly many vertices in cayley graphs generated by transposition unicyclic triangle free graphs then the resulting graphs has a large connected component that comprises of nearly all residual vertices

edge fault tolerance analysis of a class of interconnection networks fault tolerant measures have played an important role in the reliability of an interconnection network edge connectivity restricted edge connectivity extra edge connectivity and super edge connectivity of many well known interconnection networks have been explored in this paper we study the 2 extra edge connectivity of a special class of graphs g g0 g1 m proposed by chen et al appl math comput 140 2003 245 254 then by showing that several well known interconnection networks such as hyper

edge fault tolerance of interconnection networks with respect to maximally edge connectivity abstract the interconnection networks are generally modeled by a connected graph g and the connectivity of g is an important parameter for reliability and fault tolerance of the network a connected graph g v e is maximally edge connected or maximally for short if its edge connectivity attains its minimum degree we define a maximally graph g to be m maximally if g s is still maximally for any edge subset s e g with s m the maximum integer of such m denoted by m g is said to be the edge fault tolerance of g with respect to the maximally property in this paper we discuss the edge fault tolerance for maximally property of two families of interconnection networks from which we determine the exact values of m g for some well known networks

structure connectivity and substructure connectivity of hypercubes the connectivity of a network the minimum number of nodes whose removal will disconnect the network is directly related to its reliability and fault tolerability hence an important indicator of the network s robustness in this paper we extend the notion of connectivity by introducing two new kinds of connectivity called structure connectivity and substructure connectivity respectively let h be a certain particular connected subgraph of g the h structure connectivity of graph g denoted g h is the cardinality of a minimal set of subgraphs f h 1 h 2 h m in g such that every h i f is isomorphic to h and f s removal will disconnect g the h substructure connectivity of graph g denoted s g h is the cardinality of a minimal set of subgraphs f j 1 j 2 j m such that every j i f is a connected subgraph of h and f s removal will disconnect g in this paper we will establish both q n h and s q n h for the hypercube q n and h k 1 k 1 1 k 1 2 k 1 3 c 4

the h connectivity of exchanged crossed cube connectivity is an important concept in interconnection networks and reflects the fault tolerance correctly but the resilience of large networks is undervalued in this case restricted connectivity of graphs was proposed to cover the shortage the exchanged crossed cube ecq s t is a new interconnection network proposed in 2013 in this work we will show that the h connectivity of ecq s t is equal to 2h s h 1

target assignment in robotic networks distance optimality guarantees and hierarchical strategies we study the problem of multi robot target assignment to minimize the total distance traveled by the robots until they all reach an equal number of static targets in the first half of the paper we present a necessary and sufficient condition under which true distance optimality can be achieved for robots with limited communication and target sensing ranges moreover we provide an explicit non asymptotic formula for computing the number of robots needed to achieve distance optimality in terms of the robots communication and target sensing ranges with arbitrary guaranteed probabilities the same bounds are also shown to be asymptotically tight in the second half of the paper we present suboptimal strategies for use when the number of robots cannot be chosen freely assuming first that all targets are known to all robots we employ a hierarchical communication model in which robots communicate only with other robots in the same partitioned region this hierarchical communication model leads to constant approximations of true distance optimal solutions under mild assumptions we then revisit the limited communication and sensing models by combining simple rendezvous based strategies with a hierarchical communication model we obtain decentralized hierarchical strategies that achieve constant approximation ratios with respect to true distance optimality results of simulation show that the approximation ratio is as low as 1 4

k restricted edge connectivity for some interconnection networks abstract k restricted edge connectivity is an important parameter in measuring the reliability and fault tolerance of large interconnection networks in this paper we present two families of graphs similar with the networks proposed by chen et al y c chen j j m tan l h hsu s s kao super connectivity and super edge connectivity for some interconnection networks applied mathematics and computation 140 2003 245 254 and study the k k 2 3 restricted edge connectivity of these graphs in particular as the applications of our results the k k 2 3 restricted edge connectivity of the recursive circulant graphs and the n ary k cubes is given

the g extra connectivity and diagnosability of crossed cubes abstract connectivity and diagnosability are two important parameters for the fault tolerant of an interconnection network g in 1996 fabrega and fiol proposed the g extra connectivity of g in 2016 zhang et al proposed the g extra diagnosability of g that requires every component of g s has at least g 1 vertices the g extra connectivity of g is necessary for g extra diagnosability of g in this paper we show that the g extra connectivity of the crossed cube cq n is n g 1 1 2 g g 3 for n 5 0 g n 2 and the g extra diagnosability of cq n is n 1 2 g g 1 under the pmc model for n 5 0 g n 2 and the mm model for n 7 0 g n 2

extraconnectivity of k ary n cube networks given a graph g and a non negative integer g the g extraconnectivity of g is the minimum cardinality of a set of vertices in g if such a set exists whose deletion disconnects g and leaves every remaining component with more than g vertices this study shows that the 2 extraconnectivity of a k ary n cube q n k for k 4 and n 5 is equal to 6n 5

linearly many faults in dual cube like networks the dual cubes were introduced as a better interconnection network than the hypercubes for large scale distributed memory multiprocessors in this paper we introduce a generalization of these networks called dual cube like networks which preserve the basic structure of dual cubes and retain many of its topological properties we investigate structural properties of these networks beyond simple measures such as connectivity we prove that if up to kn k k 1 2 vertices are deleted from a dual cube like n regular network then the resulting graph will either be connected or will have a large component and small components having at most k 1 vertices in total and this result is sharp for k n as an application we derive additional results such as the cyclic vertex connectivity and the restricted vertex connectivity of these networks

on g extra conditional diagnosability of hierarchical cubic networks abstract connectivity and diagnosability are important parameters in measuring the fault tolerance and reliability of interconnection networks given a graph g and a non negative integer g the g extra connectivity of g denoted by g g is the minimum cardinality of a set of vertices of g if it exists whose deletion disconnects g and every remaining component has more than g vertices the g extra conditional diagnosability t g g of a graph g is the maximum value of t such that g is g extra conditional t diagnosable the n dimensional hierarchical cubic network h c n n was proposed by ghose and desai 12 as a hypercube based topology while preserving its attractive features for n 5 we first determine g h c n n for 0 g n 1 then establish t g h c n n under the pmc model for 0 g n 1 and t g h c n n under the mm model for 0 g n 1 3 respectively

vulnerability of super edge connected networks when the underlying topology of an interconnection network is modeled by a connected graph g the connectivity of g is an important measurement for reliability and fault tolerance of the network for a given integer h 0 a subset f of edges in a connected graph g is an h extra edge cut if g f is disconnected and every component has more than h vertices the h extra edge connectivity l h g of g is defined as the minimum cardinality over all h extra edge cuts of g a graph g if l h g exists is super l h if every minimum h extra edge cut of g isolates at least one connected subgraph of order h 1 the persistence r h g of a super l h graph g is the maximum integer m for which g f is still super l h for any set f e g with f m hong et al 2012 12 showed that min l 1 g d g 1 d g 1 r 0 g d g 1 where d g is the minimum vertex degree of g this paper shows that min l 2 g x g 1 d g 1 r 1 g d g 1 where x g is the minimum edge degree of g in particular for a k regular super l 1 graph g r 1 g k 1 if l 2 g does not exist or g is super l 2 and triangle free from which the exact values of r 1 g are determined for some well known networks

topological properties of twisted cube twisted cube tq n is derived by changing some connections of hypercube q n according to specific rules recently many topological properties of this variation cube are studied in this paper we prove that its connectivity is n its wide diameter and fault diameter are n 2 2 furthermore we show that tq n is a pancyclic network that is cycles of an arbitrary length at least four

super connectivity and super edge connectivity for some interconnection networks let g v e be a k regular graph with connectivity k and edge connectivity l g is maximum connected if k k and g is maximum edge connected if l k moreover g is super connected if it is a complete graph or it is maximum connected and every minimum vertex cut is x v x e for some vertex email protected v and g is super edge connected if it is maximum edge connected and every minimum edge disconnecting set is v x v x e for some vertex email protected v in this paper we present three schemes for constructing graphs that are super connected and super edge connected applying these construction schemes we can easily discuss the super connected property and the super edge connected property of hypercubes twisted cubes crossed cubes mobius cubes split stars and recursive circulant graphs

edge fault tolerance of super edge connectivity for three families of interconnection networks let g v e be a connected graph g is said to be super edge connected or super l for short if every minimum edge cut of g isolates one of the vertex of g a graph g is called m super l if for any edge set s e g with s

on g extra connectivity of folded hypercubes let g be a connected graph and g a non negative integer the g extra connectivity of g is the minimum cardinality of a set of vertices in g if it exists whose deletion disconnects g and leaves each remaining component with more than g vertices in several recent publications the g extra connectivity of an n dimensional folded hypercube was determined for g 3 and some specific n see for example chang tsai and hsieh 2014 4 in this paper an extension of the above results to all 0 g n 1 and n 7 is presented

super connectivity of k regular interconnection networks abstract super connectivity is an important issue in interconnection networks it has been shown that if a network possesses the super connectivity property it has a high reliability and a small vertex failure rate many interconnection networks like the hypercubes twisted cubes crossed cubes mobius cubes split stars and recursive circulant graphs are proven to be super connected and the augmented cubes are maximum connected however each network vertex has a higher degree as long as the number of vertices increases exponentially for example each vertex of the hypercube qn has a degree of n and each vertex of the augmented cube aqn has a degree of 2n 1 in this paper we not only show that the augmented cube aqn is super connected for n 1 2 and n 4 but also propose a variation of aqn denoted by aqn i such that v aqn i v aqn e aqn i e aqn and aqn i is i regular with n 3 and 3 i 2n 1 in which aqn i is also super connected in addition we state the diameter of aqn i

maximally local connectivity and connected components of augmented cubes abstract the connectivity of a graph is an important issue in graph theory and is also one of the most important factors in evaluating the reliability and fault tolerance of a network it is known that the augmented cube aq n is maximally connected i e 2 n 1 connected for n 4 by the classic menger s theorem every pair of vertices in aq n is connected by 2 n 1 vertex disjoint paths for n 4 a routing with parallel paths can speed up transfers of large amounts of data and increase fault tolerance motivated by research on networks with faults we obtained the result that for any faulty vertex set f v aq n and f 2 n 7 for n 4 each pair of non faulty vertices denoted by u and v in aq n f is connected by min deg f u deg f v vertex disjoint fault free paths where deg f u and deg f v are the degree of u and v in aq n f respectively moreover we demonstrate that for any faulty vertex set f v aq n and f 4 n 9 for n 4 there exists a large connected component with at least 2 n f 1 vertices in aq n f which improves on the results of ma et al 2008 who show this for n 6