solution to a conjecture on the maximal energy of bipartite bicyclic graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let c n denote the cycle of order n and p n 6 6 the graph obtained from joining two cycles c 6 by a path p n 12 with its two leaves let b n denote the class of all bipartite bicyclic graphs but not the graph r a b which is obtained from joining two cycles c a and c b a b 10 and a b 2 mod 4 by an edge in i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005 gutman and vidovic conjectured that the bicyclic graph with maximal energy is p n 6 6 for n 14 and n 16 in x li j zhang on bicyclic graphs with maximal energy linear algebra appl 427 2007 87 98 li and zhang showed that the conjecture is true for graphs in the class b n however they could not determine which of the two graphs r a b and p n 6 6 has the maximal value of energy in b furtula s radenkovic i gutman bicyclic molecular graphs with the greatest energy j serb chem soc 73 4 2008 431 433 numerical computations up to a b 50 were reported supporting the conjecture so it is still necessary to have a mathematical proof to this conjecture this paper is to show that the energy of p n 6 6 is larger than that of r a b which proves the conjecture for bipartite bicyclic graphs for non bipartite bicyclic graphs the conjecture is still open

unicyclic graphs with large energy we study the energy i e the sum of the absolute values of all eigenvalues of so called tadpole graphs which are obtained by joining a vertex of a cycle to one of the ends of a path by means of the coulson integral formula and careful estimation of the resulting integrals we prove two conjectures on the largest and second largest energy of a unicyclic graph due to caporossi cvetkovic gutman and hansen and gutman furtula and hua respectively moreover we characterise the non bipartite unicyclic graphs whose energy is largest

hermitian laplacian matrix and positive of mixed graphs a mixed graph is obtained from an undirected graph by orienting a subset of its edges the hermitian adjacency matrix of a mixed graph m of order n is the n n matrix h m h k l where h k l h l k i i 1 if there exists an orientation from vk to vl and h k l h l k 1 if there exists an edge between vk and vl but not exist any orientation and h k l 0 otherwise the value of a mixed walk w v 1 v 2 v 3 v l is h w h 12 h 23 h l 1 l a mixed walk is positive negative if h w 1 h w 1 a mixed cycle is called positive if its value is 1 a mixed graph is positive if each of its mixed cycle is positive in this work we firstly present the necessary and sufficient conditions for the positive of a mixed graph secondly we introduce the incident matrix and hermitian laplacian matrix of a mixed graph and derive some results about the hermitian laplacian spectrum

universality of 2 qudit ternary reversible gates in this paper we investigate the synthesis of ternary reversible circuits in the absence of ancilla bits we demonstrated that 2 qudit ternary swap not and 1 controlled not gates are universal for realization of arbitrary ternary n qudit reversible circuits without ancilla qudits and all even ternary n qudit reversible circuits can be constructed by ternary not and ternary 1 controlled not gates without ancilla qudits the realization approach is constructive this result is significantly different from the binary case

degree based entropies of networks revisited studies on the information content of graphs and networks have been initiated in the late fifties based on the seminal work due to shannon and rashevsky various graph parameters have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy in cao et al extremality of degree based graph entropies inform sci 278 2014 22 33 we studied graph entropies which are based on vertex degrees by using so called information functionals as a matter of fact there has been very little work to find their extremal values when considered shannon entropy based graph measures we pursue with this line of research by proving further extremal properties of the degree based graph entropies

bicyclic signed digraphs with maximal energy abstract the energy of a digraph sidigraph is defined as the sum of absolute values of real parts of its eigenvalues monsalve and rada 2016 determined the bicyclic digraphs with maximal energy among the set of all bicyclic digraphs with fixed order in this paper we find the bicyclic sidigraphs with maximal energy over the set of all bicyclic sidigraphs with fixed order we also construct few families of non cospectral equienergetic bicyclic sidigraphs

expander graphs and their applications a major consideration we had in writing this survey was to make it accessible to mathematicians as well as to computer scientists since expander graphs the protagonists of our story come up in numerous and often surprising contexts in both fields but perhaps we should start with a few words about graphs in general they are of course one of the prime objects of study in discrete mathematics however graphs are among the most ubiquitous models of both natural and human made structures in the natural and social sciences they model relations among species societies companies etc in computer science they represent networks of communication data organization computational devices as well as the flow of computation and more in mathematics cayley graphs are useful in group theory graphs carry a natural metric and are therefore useful in geometry and though they are just one dimensional complexes they are useful in certain parts of topology e g knot theory in statistical physics graphs can represent local connections between interacting parts of a system as well as the dynamics of a physical process on such systems the study of these models calls then for the comprehension of the significant structural properties of the relevant graphs but are there nontrivial structural properties which are universally important expansion of a graph requires that it is simultaneously sparse and highly connected expander graphs were first defined by bassalygo and pinsker and their existence first proved by pinsker in the early 70s the property of being an expander seems significant in many of these mathematical computational and physical contexts it is not surprising that expanders are useful in the design and analysis of communication networks what is less obvious is that expanders have surprising utility in other computational settings such as in the theory of error correcting codes and the theory of pseudorandomness in mathematics we will encounter e g their role in the study of metric embeddings and in particular in work around the baum connes conjecture expansion is closely related to the convergence rates of markov chains and so they play a key role in the study of monte carlo algorithms in statistical mechanics and in a host of practical computational applications the list of such interesting and fruitful connections goes on and on with so many applications we will not even

unicyclic graphs with maximal energy abstract let 1 2 n be the eigenvalues of a graph g of order n the energy of g is defined as e g 1 2 n let p n 6 6 be the graph obtained from two copies of c 6 joined by a path p n 10 b n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles c a and c b a b 10 and a b 2 mod 4 joined by an edge in this paper we show that p n 6 6 is the graph with maximal energy in b n which gives a partial solution to gutman s conjecture in gutman and vidovic 2001 i gutman d vidovic quest for molecular graphs with maximal energy a computer experiment j chem inf sci 41 2001 1002 1005

extremal laplacian energy of directed trees unicyclic digraphs and bicyclic digraphs abstract let a g and d g be the adjacency matrix of a digraph g with n vertices and the diagonal matrix of vertex outdegrees of g respectively then the laplacian matrix of the digraph g is l g d g a g the laplacian energy of a digraph g is defined as l e g i 1 n i 2 by using second spectral moment where 1 2 n are all the eigenvalues of l g of g in this paper by using arc shifting operation and out star shifting operation we determine the directed trees unicyclic digraphs and bicyclic digraphs which attain maximal and minimal laplacian energy among all digraphs with n vertices respectively

solutions to unsolved problems on the minimal energies of two classes of trees the energy of a graph is defined as the sum of the absolute values of all eigenvalues of the graph let t n p t n d be the set of all trees of order n with p pendent vertices diameter d respectively in this paper we completely characterize the trees with second minimal and third minimal energy in t n p t n d respectively for 4 p n 9 10 d n 3 respectively which solves the problems left in ma 2014

the general connectivity indices of fluoranthene type benzenoid systems the general connectivity index r g of a graph g v e is defined as r g u v e g d u d v where u v is an edge of g du and dv denote the degrees of the vertices u and v respectively and 0 is an arbitrary real number in this paper we give the expressions for computing the general connectivity indices of fluoranthene type benzenoid systems and we determine the extremal values of r g in f benzenoid systems with h hexagons for some real number especially the extremal values of r g in cata catacondensed fluoranthene type benzenoid systems with h hexagons were studied

real space renormalization group approach to driven diffusive systems we introduce a real space renormalization group procedure for driven diffusive systems which predicts both steady state and dynamic properties we apply the method to the boundary driven asymmetric simple exclusion process and recover exact results for the steady state phase diagram as well as the crossovers in the relaxation dynamics for each phase

extremality of degree based graph entropies abstract many graph invariants have been used for the construction of entropy based measures to characterize the structure of complex networks based on shannon s entropy we study graph entropies which are based on vertex degrees by using so called information functionals when considering shannon entropy based graph measures there has been very little work to find their extremal values the main contribution of this paper is to prove some extremal values for the underlying graph entropy of certain families of graphs and to find the connection between the graph entropy and the sum of degree powers further conjectures to determine extremal values of graph entropies are given

bicyclic oriented graphs with skew rank 6 let g be an oriented graph and s g be its skew adjacency matrix the skew rank of g denoted by sr g is the rank of s g in this paper we characterize all the bicyclic oriented graphs with skew rank 6 let g be a bicyclic oriented graph with pendant vertices but no pendant twins if s r g 6 then 6 v g 10

sharp bounds on the distance spectral radius and the distance energy of graphs the d eigenvalues 1 2 p of a graph g are the eigenvalues of its distance matrix d and form the d spectrum of g denoted by specd g the greatest d eigenvalue is called the d spectral radius of g denoted by 1 the d energy ed g of the graph g is the sum of the absolute values of its d eigenvalues in this paper we obtain some lower bounds for 1 and characterize those graphs for which these bounds are best possible we also obtain an upperbound for ed g and determine those maximal d energy graphs

on the maximal energy tree with two maximum degree vertices abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix for 3 and t 3 denote by t a t or simply t a the tree formed from a path p t on t vertices by attaching 1 p 2 s on each end of the path p t and t b t or simply t b the tree formed from p t 2 by attaching 1 p 2 s on an end of the p t 2 and 2 p 2 s on the vertex next to the end in li et al 2009 16 proved that among trees of order n with two vertices of maximum degree the maximal energy tree is either the graph t a or the graph t b where t n 4 4 3 however they could not determine which one of t a and t b is the maximal energy tree this is because the quasi order method is invalid for comparing their energies in this paper we use a new method to determine the maximal energy tree it turns out that things are more complicated we prove that the maximal energy tree is t b for 7 and any t 3 while the maximal energy tree is t a for 3 and any t 3 moreover for 4 the maximal energy tree is t a for all t 3 but one exception that t 4 for which t b is the maximal energy tree for 5 the maximal energy tree is t b for all t 3 but 44 exceptions that t is both odd and 3 t 89 for which t a is the maximal energy tree for 6 the maximal energy tree is t b for all t 3 but three exceptions that t 3 5 7 for which t a is the maximal energy tree one can see that for most cases of t b is the maximal energy tree 5 is a turning point and 3 and 4 are exceptional cases which means that for all chemical trees whose maximum degrees are at most 4 with two vertices of maximum degree at least 3 t a has maximal energy with only one exception t a 4 4

multiple valued logic its status and its future multiple valued logic in which the number of discrete logic levels is not confined to two has been the subject of much research over many years the practical objective of this work has been to increase the information content of the digital signals in a system to a higher value than that provided by binary operation in this tutorial survey paper we will review the historical developments in this field both in circuit realizations and in methods of handling multiple valued design data and consider the present state of the art and future expectations

bicyclic digraphs with maximal energy if d is a digraph with n vertices then the energy of d is defined as e d k 1 n re z k where re z 1 re zn are the real parts of the eigenvalues z 1 z n of d in this paper we solve a problem proposed in khan et al 2015 we find the maximal value of the energy over the set of all bicyclic digraphs b n with n vertices

on the tree with diameter 4 and maximal energy the energy of a simple graph is defined as the sum of the absolute values of all the eigenvalues of its adjacency matrix jianping found the structure of trees which have maximal energy among all trees of diameter 4 whose centers are of degree t this paper generalizes the result we determine the structure of the general tree of diameter 4 and maximal energy

bicyclic signed graphs with minimal and second minimal energy abstract let s g be a signed graph of order n and size m where g is its underlying graph and is the sign function a connected signed graph s is said to be bicyclic if m n 1 in this paper we provide characterizations of bicyclic signed graphs with minimal and second minimal energy we notice that there are gaps in the proofs of the main results in j zhang h kan 2014 32 and j zhang b zhou 2005 31 which deal with minimal energy of bicyclic graphs and we plug these gaps as a consequence of the results obtained for bicyclic signed graphs

block transitive 2 v k 1 designs and the chevalley groups f 4 q this paper is a contribution to the study of the automorphism groups of 2 v k 1 designs our aim is to classify pairs d g in which d is a 2 v k 1 design and g is a block transitive group of automorphisms of d it is clear that if one wishes to study the structure of a finite group acting on a 2 v k 1 design then describing the socle is an important first step let g act as a block transitive and point primitive automorphism group of a 2 v k 1 design d set k 2 k v 1 in this paper we prove that when q p a for some prime power and q is large specifically q 2 2 k 2 k k 2 1 a then the socle of g is not f 4 q

majority based reversible logic gates reversible logic plays an important role in the synthesis of circuits for quantum computing in this paper we introduce families of reversible gates based on the majority boolean function mbf and we prove their properties in reversible circuit synthesis these gates can be used to synthesize reversible circuits of minimum scratchpad register width for arbitrary reversible functions we show that given a mbf f with 2k 1 inputs f can be implemented by a reversible logic gate with 2k 1 inputs and 2k 1 outputs i e without any constant inputs we also demonstrate new gates from this family with very efficient quantum realizations for majority based applications they can be used to synthesize any reversible function of the same width in conjunction with inverters and feynman 2 qubit controlled not gates the gate universality problem is formulated in terms of elementary group theory and solved using the algebraic software gap

bounds for the sum balaban index and revised szeged index of regular graphs mathematical properties of many topological indices are investigated knor et al gave an upper bound for the balaban index of r regular graphs on n vertices and a better upper bound for fullerene graphs they also suggested exploring similar bounds for other topological indices in this paper we consider the sum balaban index and the revised szeged index and give upper and lower bounds for these three indices of r regular graphs and also the cubic graphs and fullerene graphs respectively

complete solution to a problem on the maximal energy of unicyclic bipartite graphs abstract the energy of a simple graph g denoted by e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix denote by c n the cycle and p n 6 the unicyclic graph obtained by connecting a vertex of c 6 with a leaf of p n 6 caporossi et al conjectured that the unicyclic graph with maximal energy is p n 6 for n 8 12 14 and n 16 in hou et al 2002 y hou i gutman c woo unicyclic graphs with maximal energy linear algebra appl 356 2002 27 36 the authors proved that e p n 6 is maximal within the class of the unicyclic bipartite n vertex graphs differing from c n and they also claimed that the energies of c n and p n 6 is quasi order incomparable and left this as an open problem in this paper by utilizing the coulson integral formula and some knowledge of real analysis especially by employing certain combinatorial techniques we show that the energy of p n 6 is greater than that of c n for n 8 12 14 and n 16 which completely solves this open problem and partially solves the above conjecture

an edge grafting theorem on the energy of unicyclic and bipartite graphs the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix the edge grafting operation on a graph is certain kind of edge moving between two pendant paths starting from the same vertex in this paper we show how the graph energy changes under the edge grafting operations on unicyclic and bipartite graphs we also give some applications of this result on the comparison of graph energies between unicyclic or bipartite graphs

two stage coarsening mechanism in a kinetically constrained model of an attractive colloid we study an attractive version of the east model using the real space renormalization group rg introduced by stella et al the former is a kinetically constrained model with an ising like interaction between excitations and shows striking agreement with the phenomonology of attractive colloidal systems we find that the rg predicts two nonuniversal dynamic exponents which suggests that in the out of equilibrium regime the model coarsens via a two stage mechanism we explain this mechanism physically and verify this prediction numerically in addition we predict that the characteristic relaxation time of the model is a nonmonotonic function of attraction strength again in agreement with numerical results

extremal laplacian energy of threshold graphs let g be a connected threshold graph of order n with m edges and trace t in this paper we give a lower bound on laplacian energy in terms of n m and t of g from this we determine the threshold graphs with the first four minimal laplacian energies moreover we obtain the threshold graphs with the largest and the second largest laplacian energies

spectral properties of geometric arithmetic index the concept of geometric arithmetic index was introduced in the chemical graph theory recently but it has shown to be useful one of the main aims of algebraic graph theory is to determine how or whether properties of graphs are reflected in the algebraic properties of some matrices the aim of this paper is to study the geometric arithmetic index ga1 from an algebraic viewpoint since this index is related to the degree of the vertices of the graph our main tool will be an appropriate matrix that is a modification of the classical adjacency matrix involving the degrees of the vertices moreover using this matrix we define a ga laplacian matrix which determines the geometric arithmetic index of a graph and satisfies properties similar to the ones of the classical laplacian matrix

the energy of directed hexagonal systems the energy of a digraph d is defined as e d i 1n re zi where re zi denotes the real part of the complex number zi we study in this work the energy over the set n consisting of digraphs with n vertices and cycles of length 2mod 4 due to the fact that the characteristic polynomial of a digraph d n has an expression of the form d z zn k 1 n2 1 kc2k d zn 2k where c2k d are nonnegative integers it is possible to define a quasi order relation over n in such a way that the energy is increasing moreover we show that the energy of a digraph d n decreases when an arc of a cycle of length 2 is deleted consequently we obtain extremal values of the energy over sets of directed hexagonal systems i e digraphs whose underlying graph is a hexagonal system

spectral norm of oriented graphs abstract we show that the spectral radius of a graph is an upper bound on the spectral norm of any of its orientations we also show that equality is attained only when the orientation is sink source as an application we find the extremal values of the spectral norm over the set of oriented trees oriented unicyclic graphs and connected digraphs with n vertices and n arcs

realizing ternary quantum switching networks without ancilla bits this paper investigates the synthesis of quantum networks built to realize ternary switching circuits in the absence of ancilla bits the results we established are twofold the first shows that ternary swap ternary not and ternary toffoli gates are universal for the realization of arbitrary n n ternary quantum switching networks without ancilla bits the second result proves that all n n quantum ternary networks can be generated by not controlled not multiply two and toffoli gates our approach is constructive

minimal energies of trees with given parameters abstract the energy of a graph is the sum of the absolute values of the eigenvalues of its adjacency matrix let t k be the set of trees with given order k suppose that t t k and v 1 v 2 v k be the ordering vertex set of t we denote by t n 1 n 2 n k the graph obtained by attaching n i pendent vertices to vertex v i i 1 2 k of t respectively let t n k t n 1 n 2 n k t t k n 1 n 2 n k n k n i 1 i 1 2 k in this paper we determine the trees in t n k with the first and the second minimal energies as applications we can characterize the trees with the first and the second minimal energies among the set of trees with given domination number matching number independence number respectively

oriented unicyclic graphs with the first n 92 largest skew energies abstract let g be an oriented graph obtained by assigning an orientation to the edge set of a simple undirected graph g such that g becomes a directed graph let s g be the skew adjacency matrix of g the skew energy of g is defined as the sum of the absolute values of all eigenvalues of s g in this paper we provide a new method to compare the skew energies of two oriented graphs whose skew characteristic polynomials satisfy a given recurrence relation and determine the oriented unicyclic graphs of order n with the first n 9 2 largest skew energies for n 31

strategies for designing geometric transformations on quantum images three design strategies for constructing new geometric transformations on quantum images from other transformations are proposed the strategies focus on the affected areas in the images the separability and smoothness of the transformations by exploiting a representation of images on quantum computers extensively the complexity in terms of the number of basic gates and the depth of the corresponding circuits are analyzed to understand the advantages and disadvantages of each strategy examples to demonstrate the applicability of the proposed strategies are presented the strategies provide high level tools to explore and analyze transformations which are necessary to build practical image processing applications on quantum computers

the spectral radius of edge chromatic critical graphs abstract a connected graph g with maximum degree and edge chromatic number g 1 is called critical if g e for every edge e of g in this paper we consider two weaker versions of vizing s conjecture which concern the spectral radius g and the signless laplacian spectral radius g of g we obtain some lower bounds for g and g and present some cases where the conjectures are true finally several open problems are also proposed

the extreme eigenvalues and maximum degree of k connected irregular graphs abstract let 1 g be the largest eigenvalue and n g be the smallest eigenvalue of a k connected irregular graph g with n vertices m edges and maximum degree in this paper we prove that 1 g n 2 m k 2 n 2 m n 2 2 n k n k 2 this inequality improves previous results of several authors and implies two lower bounds on n g which also refine some known bounds another lower bound on 1 g for a connected irregular graph g is given as well

graph distance measures based on topological indices revisited graph distance measures based on topological indices have been already explored by dehmer et al also inequalities for those graph distance measures have been proved in this paper we continue studying such comparative graph measures based on the well known wiener index graph energy and randic index respectively we prove extremal properties of the graph distance measures for some special classes of graphs to demonstrate useful properties of the measures we also discuss numerical results to conclude the paper we state some open problems

spectral radius of graphs with given matching number abstract in this paper we show that of all graphs of order n with matching number the graphs with maximal spectral radius are kn if n 2 or 2 1 k 2 1 k n 2 1 if 2 2 n k k n or k 2 1 k n 2 1 if n 3 2 k k n if n 3 2 where k t is the empty graph on t vertices

the bipartite unicyclic graphs with the first n 3 4 largest matching energies the theory of matching energy of graphs since been proposed by gutman and wagner in 2012 has attracted more and more attention it is denoted by b n m the class of bipartite graphs with order n and size m in particular b n n denotes the set of bipartite unicyclic graphs which is an interesting class of graphs in this paper for odd n we characterize the bipartite unicyclic graphs with the first n 3 4 largest matching energies there is an interesting correspondence we conclude that the graph with the second maximal matching energy in b n n for odd n 11 is p n 6 which is the only graph attaining the maximum value of the energy among all the bipartite unicyclic graphs for n 16

on tricyclic graphs of a given diameter with minimal energy abstract the energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph let g n d be the class of tricyclic graphs g on n vertices with diameter d and containing no vertex disjoint odd cycles c p c q of lengths p and q with p q 2 mod 4 in this paper we characterize the graphs with minimal energy in g n d

implementing an arbitrary reversible logic gate the 2w reversible logic gates of width w i e reversible logic gates with w inputs and w outputs together with the action of cascading form a group r we define a subgroup k consisting of the switched gates there are 2w 1 2 such gates they partition the group r into 2w 1 1 double cosets this allows us to decompose an arbitrary reversible gate into the cascade of three gates a switched gate an upside down simple control gate and a second switched gate we present an algorithm to perform this factorization and thus provide a method of implementing an arbitrary reversible gate into hardware the algorithm can be used to automate the implementation of a reversible function in future c mos technologies realizing low cost computing

a bound on the spectral radius of graphs abstract let g be a simple connected graph with n vertices and m edges let a be the adjacency matrix of g the spectral radius a of a satisfies a 2m n 1 with equality if and only if g is isomorphic to one of the following two graphs a the star k1 n 1 b the complete graph kn

bicyclic oriented graphs with skew rank 2 or 4 the skew rank of oriented graph g denoted by sr g is the rank of the skew adjacency matrix of g the skew rank is even since the skew adjacency matrix is skew symmetric in this paper we characterize the bicyclic oriented graphs with skew rank 2 or 4

the energy of a graph abstract the energy e g of a simple graph g is defined to be the sum of the absolute values of the eigen values of g if g is a k regular graph on n vertices then e g k k n 1 n k b 2 and this bound is sharp it is shown that for each 0 there exist infinitely many n for each of which there exists a k regular graph g of order n with k e g b 2 two graphs with the same number of vertices are equienergetic if they have the same energy we show that for any positive integer n 3 there exist two equienergetic graphs of order 4n that are not cospectral

quantum scaling approach to nonequilibrium models stochastic nonequilibrium exclusion models are treated using a real space scaling approach the method exploits the mapping between nonequilibrium and quantum systems and it is developed to accommodate conservation laws and duality symmetries yielding exact fixed points for a variety of exclusion models in addition it is shown how the asymmetric simple exclusion process in one dimension can be written in terms of a classical hamiltonian in two dimensions using a suzuki trotter decomposition

on the minimal energy of graphs abstract the energy of a graph is the sum of the absolute values of the eigenvalues of the graph which is used to approximate the total electron energy of the molecule in this paper we determine the n e graphs with minimal energy for e n 1 and n 2 which is giving a complete solution to the conjecture for e n 1 and e n 2 proposed by caporossi et al in 4 moreover we determine the graphs with the minimal and second minimal energies for n 1 e 3 n 2 3 and the unique graph with minimal energy for 3 n 5 2 e 2 n 4 among all quasi trees with n vertices and e edges respectively

extremal iota energy of bicyclic digraphs the energy of an n vertex digraph d is defined by e d k 1n re zk where z1 zn are eigenvalues of d and re zk is the real part of eigenvalue zk very recently a new type of energy of digraphs has been introduced which is known as the iota energy of digraphs the iota energy of the digraph d is defined by ec d k 1n im zk where z1 zn are eigenvalues of d and im zk is the imaginary part of eigenvalue zk the unicyclic digraphs with extremal iota energy are known in this paper we consider a class dn of n vertex bicyclic digraphs with vertex disjoint directed cycles and find the digraphs in dn with minimal and maximal iota energy

generating the group of reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to have an insight into the structure of the group formed by the reversible logic gates and their cascading into reversible circuits such insight is gained from constructing chains of maximal subgroups the subgroup of control gates plays a prominent role as it is a sylow 2 subgroup

a computational approach to construct a multivariate complete graph invariant in this paper we present a computational approach for finding complete graph invariants specifically we generate exhaustive sets of connected non isomorphic graphs with 9 and 10 vertices and demonstrate that a 97 dimensional multivariate graph invariant is capable to distinguish each of the non isomorphic graphs furthermore in order to tame the computational complexity of the problem caused by the vast number of graphs e g involving over 10 million networks with 10 vertices we suggest a low dimensional iterative procedure that is based on highly discriminative individual graph invariants we show that also this computational approach leads to a perfect discrimination overall our numerical results prove the existence of such graph invariants for networks with 9 and 10 vertices furthermore we show that our iterative approach has a polynomial time complexity

the decomposition of an arbitrary reversible logic circuit the 2w reversible logic circuits of width w i e reversible logic circuits with w inputs and w outputs together with the action of cascading form a group g isomorphic to the symmetric group we define two conjugate subgroups g1 and g2 together they partition the group g into 2w 1 1 double cosets these allow us to decompose an arbitrary member of g into a cascade of three simpler members this decomposition is a far relative of the well known lu decomposition of a square matrix

a note on the spectral characterization of strongly connected bicyclic digraphs let d be a digraph with vertex set v d and a be the adjacency matrix of d in this paper we characterize the extremal digraphs which achieve the maximum and minimum spectral radius among strongly connected bicyclic digraphs furthermore we show that any strongly connected bicyclic digraph is determined by the spectrum

extremal values of matching energies of one class of graphs in 1978 gutman proposed the concept of graph energy defined as the sum of the absolute values of eigenvalues of the adjacency matrix of a molecular graph which is related to the energy of electrons in conjugated hydrocarbons recently gutman and wagner proposed the concept of matching energy and pointed out that the chemical applications of matching energy go back to the 1970s in this paper we study the extremal values of the matching energy and characterize the graphs with minimal matching energy among all tricyclic graphs with a given diameter our methods can help to find more extremal values for other classes of molecular networks and the results suggest the structures with extremal energies

on the signless laplacian spectral radius of irregular graphs abstract let g be a connected irregular graph in this paper we give lower and upper bounds for the signless laplacian radius q 1 of g we show that q 1 4 e n 2 2 n and q 1 2 1 n d 1 4 hold where d is the diameter of g

r universal reversible logic gates reversible logic plays a fundamental role both in ultra low power electronics and in quantum computing it is therefore important to know which reversible logic gates can be used as building block for the reversible implementation of an arbitrary boolean function and which cannot

energy of matrices abstract let m n c denote the space of n n matrices with entries in c we define the energy of a m n c as 1 e a k 1 n k t r a n where 1 n are the eigenvalues of a tr a is the trace of a and z denotes the modulus of z c if a is the adjacency matrix of a graph g then e a is precisely the energy of the graph g introduced by gutman in 1978 in this paper we compare the energy e with other well known energies defined over matrices then we find upper and lower bounds of e which extend well known results for the energies of graphs and digraphs also we obtain new results on energies defined over the adjacency laplacian and signless laplacian matrices of digraphs

on the spectral radius of graphs with a given domination number we characterize the graphs which achieve the maximum value of the spectral radius of the adjacency matrix in the sets of all graphs with a given domination number and graphs with no isolated vertices and a given domination number

the spectral radius of irregular graphs abstract let 1 be the largest eigenvalue and n the least eigenvalue of the adjacency matrix of a connected graph g of order n we prove that if g is irregular with diameter d maximum degree minimum degree and average degree d then 1 n d 1 d d 2 1 the inequality improves previous bounds of various authors and implies two lower bounds on n which improve previous bounds of nikiforov it also gives some fine tuning of a result of alon and sudakov a similar inequality is also obtained for the laplacian spectral radius of a connected irregular graph

realization and synthesis of reversible functions reversible circuits play an important role in quantum computing this paper studies the realization problem of reversible circuits for any n bit reversible function we present a constructive synthesis algorithm given any n bit reversible function there are n distinct input patterns different from their corresponding outputs where email protected 2 n and the other 2 n n input patterns will be the same as their outputs we show that this circuit can be synthesized by at most email protected n n 1 cnot gates and 4n email protected n not gates the time and space complexities of the algorithm are w email protected 4 n and w email protected 2 n respectively the computational complexity of our synthesis algorithm is exponentially lower than that of breadth first search based synthesis algorithms

note on unicyclic graphs with given number of pendent vertices and minimal energy abstract for a simple graph g the energy e g is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix let g n p denote the set of unicyclic graphs with n vertices and p pendent vertices in h hua m wang unicyclic graphs with given number of pendent vertices and minimal energy linear algebra appl 426 2007 478 489 hua and wang discussed the graphs that have minimal energy in g n p and determined the minimal energy graphs among almost all different cases of n and p in their work certain case of the values was left as an open problem in which the minimal energy species have to be chosen in two candidate graphs but cannot be determined by comparing of the corresponding coefficients of their characteristic polynomials this paper aims at solving the problem completely by using the well known coulson integral formula

on the symmetry of logic embedding logical operations in non dissipative physical processes requires the use of reversible logic following feynman s 1986 approach the sixteen distinct truth tables of classical logic are shown to be contained in the 8 reversible logic operations covered by the symmetric group s8 which permute the eight values of three logical variables small subgroups of s8 are shown to cover respectively reversible logic reversible switching and reversible arithmetic a new universal primitive is found which generates a covering group of reversible logic it is shown that the octahedral group in four dimensions covers both reversible logic and switching and hence that the orthogonal group o 4 provides a covering group for quantum gates

on unicyclic graphs with given number of pendent vertices and minimal energy abstract let g n p denote the set of unicyclic graphs with n vertices and p 1 pendent vertices hua and wang 2007 4 claim to have determined the graph with minimal energy in g n p except for the case when n p 5 which was left as an open problem this case was later solved by huo et al 2010 5 in this paper we provide a counter example to show that the main result of hua and wang 2007 is not true we also give the correct form of the characterization of unicyclic graphs with given number of pendent vertices and minimal energy we make use of the well known coulson s integral formula for the energy of graphs our proof is more general and also includes the case n p 5 considered by huo ji and li 2010

a classification of the maximal subgroups of the finite alternating and symmetric groups following the classification of finite simple groups one of the major problems in finite group theory today is the determination of the maximal subgroups of the almost simple groups that is of groups x such that x0 u x aut x for some finite non abelian simple group x0 the problem has been solved for most sporadic groups and groups of lie type of low rank see 19 sects 3 43 for discussion and references this paper is a contribution to the case where x0 is an alternating group a so that for n 6 x is a or s the maximal subgroups of a and s are known for several classes of degrees n

on ordering of complements of graphs with respect to matching numbers we consider an earlier much studied quasi order defined in terms of matching numbers of graphs and apply it to graph complements we establish four transformations on the complements of graphs that increase or decrease the matching numbers accordingly several applications of these results are put forward

spectral radius and signless laplacian spectral radius of strongly connected digraphs abstract let d be a strongly connected digraph and a d be the adjacency matrix of d let diag d be the diagonal matrix with outdegrees of the vertices of d and q d diag d a d be the signless laplacian matrix of d the spectral radius of q d is called the signless laplacian spectral radius of d denoted by q d in this paper we give a sharp bound on q d where d has a given outdegree sequence and compare the bound with known bounds we establish some sharp upper or lower bound on q d with some given parameter such as clique number girth or vertex connectivity and characterize the extremal graph in addition we also determine the unique digraph which achieves the minimum or maximum the second minimum or maximum the third minimum the fourth minimum spectral radius and signless laplacian spectral radius among all strongly connected digraphs and answer the open problem proposed by lin and shu 14

some properties of graphs constructed from 2 designs abstract let d p b be any nontrivial nonsymmetric 2 v k design in this paper we construct a connected regular and bipartite graph say ig v b d from d some graphic properties of ig v b d are investigated in particular the results in the paper bridges the equivalency between edge transitivity of ig v b d and flag transitivity of d with the equivalency it is obtained that ig v b d is a semisymmetric graph as long as d is a nonsymmetric flag transitive 2 v k design