maximizing the least signless laplacian eigenvalue of unicyclic graphs abstract in this paper a graph with the maximum least signless laplacian eigenvalue among all connected unicyclic graphs with fixed order is determined

the smallest eigenvalue of the signless laplacian recently the signless laplacian matrix of graphs has been intensively investigated while there are many results about the largest eigenvalue of the signless laplacian the properties of its smallest eigenvalue are less well studied the present paper surveys the known results and presents some new ones about the smallest eigenvalue of the signless laplacian

a note on the positive semidefiniteness of aα g abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for every real 0 1 write a g for the matrix a g d g 1 a g let 0 g be the smallest for which a g is positive semidefinite it is known that 0 g 1 2 the main results of this paper are 1 if g is d regular then 0 min a g d min a g where min a g is the smallest eigenvalue of a g 2 g contains a bipartite component if and only if 0 g 1 2 3 if g is r colorable then 0 g 1 r

dynamical properties of a randomly diluted neural network with variable activity the subject of this paper is a neural network with binary neurons randomly diluted synapses and variable pattern activity we look at the system with parallel updating using a probabilistic approach to solve the one step dynamics with one condensed pattern we derive restrictions on the storage capacity and the mutual information content occuring during the retrieval process special focus is placed on the constraints on the threshold for optimal performance we also look at the effect of noisy updating giving a dynamical version of the critical temperature the corresponding threshold and an approximation for the time evolution for small temperatures the description is applicable to the whole retrieval process in the limit of strong dilution the analysis is carried out as exactly as possible and over the whole parameter ranges generalizing some former results

signless laplacian energy of a graph and energy of a line graph abstract for a simple graph g of order n size m and with signless laplacian eigenvalues q 1 q 2 q n the signless laplacian energy q e g is defined as q e g i 1 n q i d where d 2 m n is the average vertex degree of g we obtain the lower bounds for q e g in terms of first zagreb index m 1 g maximum degree d 1 second maximum degree d 2 minimum degree d n and second minimum degree d n 1 as a consequence of these bounds we obtain several bounds for the energy e l g of the line graph l g of graph g in terms of various graph parameters like m 1 g the clique number n m etc which improve some recently known bounds

the least eigenvalue of signless laplacian of graphs under perturbation abstract we investigate how the least eigenvalue of the signless laplacian of a graph changes by relocating a bipartite branch from one vertex to another vertex and minimize the least eigenvalue of the signless laplacian among the class of connected graphs with fixed order which contains a given non bipartite graph as an induced subgraph

the three state layered neural network with finite dilution abstract the dynamics and the stationary states of an exactly solvable three state layered feed forward neural network model with asymmetric synaptic connections finite dilution and low pattern activity are studied in extension of a recent work on a recurrent network detailed phase diagrams are obtained for the stationary states and for the time evolution of the retrieval overlap with a single pattern it is shown that in spite of instabilities for low thresholds there is a gradual improvement in network performance with increasing threshold up to an optimal stage the robustness to synaptic noise is checked and the effects of dilution and of variable threshold on the information content of the network are also established

positive solutions for a quasilinear elliptic equation of kirchhoff type this paper is concerned with the existence of positive solutions to the class of nonlocal boundary value problems of the type m w u 2dx du f x u email protected u 0 email protected email protected where w is a smooth bounded domain of n m is a positive function and f has subcritical growth

on the parallel dynamics of the q state potts and q ising neural networks using a probabilistic approach the parallel dynamics of theq state potts andq ising neural networks are studied at zero and at nonzero temperatures evolution equations are derived for the first time step and arbitraryq these formulas constitute recursion relations for the exact parallel dynamics of the extremely diluted asymmetric versions of these networks an explicit analysis including dynamical capacity temperature diagrams and the temperature dependence of the overlap is carried out forq 3 both types of models are compared

on the aα spectra of graphs abstract let g be a graph with adjacency matrix a g and let d g be the diagonal matrix of the degrees of g for any real 0 1 nikiforov 8 defined the matrix a g as a g d g 1 a g in this paper we give some results on the eigenvalues of a g for 1 2 in particular we characterize the graphs with k a g n 1 for 2 k n moreover we show that n a g 2 1 if g contains no isolated vertices

the domination number and the least q eigenvalue abstract in this paper for a nonbipartite graph with both order n and domination number we show that it contains a nonbipartite unicyclic spanning subgraph with the same domination number we also present some results about the changing of the domination number when the structure of a graph is changed by investigating the relation between the domination number and the least q eigenvalue of a graph we minimize the least q eigenvalue among all the nonbipartite graphs with both order n 4 and given domination number n 1 3

on the least eigenvalue of aα matrix of graphs abstract let g be a graph of order n with m edges and chromatic number let a g be the adjacency matrix and d g be the diagonal matrix of vertex degrees of g nikiforov defined the matrix a g as a g d g 1 a g where 0 1 then a 1 2 g 1 2 d g a g 1 2 q g where q g is the signless laplacian matrix of g let q n g and n a be the least eigenvalue of q g and a g respectively lima et al 2011 8 proposed some open problems on q n g two of which are as follows 1 to characterize the graphs for which q n g 2 m n 1 2 to characterize the graphs for which q n g 1 1 1 2 m n in this paper we present an upper bound on n a in terms of n m and 1 2 1 and characterize the extremal graphs as an application we completely solve problem 1 moreover we examine the more generalized result of problem 2 on a g when 1 we obtain some necessary conditions for n a 1 2 m 1 n and as a corollary for the equality in problem 2

existence results of positive solutions of kirchhoff type problems abstract in the present paper we use variational methods to prove two existence results of positive solutions of the following kirchhoff type problems a b u 2 u f x u in u 0 on one deals with the asymptotic behaviors of f near zero and infinity and the other deals with 4 superlinear of f at infinity

a sharp lower bound for the least eigenvalue of the signless laplacian of a non bipartite graph we prove that the minimum value of the least eigenvalue of the signless laplacian of a connected non bipartite graph with a prescribed number of vertices is attained solely in the unicyclic graph obtained from a triangle by attaching a path at one of its endvertices

the space of interactions in neural network models the typical fraction of the space of interactions between each pair of n ising spins which solve the problem of storing a given set of p random patterns as n bit spin configurations is considered the volume is calculated explicitly as a function of the storage ratio alpha p n of the value kappa 0 of the product of the spin and the magnetic field at each site and of the magnetisation m here m may vary between 0 no correlation and 1 completely correlated the capacity increases with the correlation between patterns from alpha 2 for correlated patterns with kappa 0 and tends to infinity as m tends to 1 the calculations use a saddle point method and the order parameters at the saddle point are assumed to be replica symmetric this solution is shown to be locally stable a local iterative learning algorithm for updating the interactions is given which will converge to a solution of given kappa provided such solutions exist

bounds for the largest and the smallest aα eigenvalues of a graph in terms of vertex degrees abstract let g be a graph with adjacency matrix a g and with d g the diagonal matrix of its vertex degrees nikiforov defined the matrix a g with 0 1 as a g d g 1 a g the largest and the smallest eigenvalues of a g are respectively denoted by g and n g in this paper we present a tight upper bound for g in terms of the vertex degrees of g for 1 for any graph g g max u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected for 0 1 the equality holds if and only if g is regular or bipartite semi regular as an application we solve the problem raised by nikiforov in 15 for the smallest eigenvalue n g of a bipartite graph g of order n with no isolated vertices for 0 1 then n g min u w d u d w 2 d u d w 2 4 1 2 d u d w 2 if g is connected and 1 2 the equality holds if and only if g is regular or semi regular

on the least distance eigenvalue of a graph abstract denote by d g d i j n n the distance matrix of a connected graph g with n vertices where d i j is equal to the distance between vertices v i and v j in g the least eigenvalue of d g is called the least distance eigenvalue of g denoted by n in this paper we determine all the graphs with n 2 383 0

sign changing and multiple solutions of kirchhoff type problems without the p s condition abstract the existence of nontrivial solutions of kirchhoff type equations is an important nonlocal quasilinear problem in this paper we use minimax methods and invariant sets of descent flow to prove two interesting existence theorems for the following 4 superlinear kirchhoff type problems without the p s condition one concerning the existence of a nontrivial solution and the other one concerning the existence of sign changing solutions and multiple solutions a b u 2 u f x u in u 0 on

the least q eigenvalue with fixed domination number abstract denote by lg l the lollipop graph obtained by attaching a pendant path p v g v g 1 v g l l 1 to a cycle c v 1 v 2 v g v 1 g 3 a f g l g r a p h of order n g 1 is defined to be the graph obtained by attaching n g l pendent vertices to some of the nonpendant vertices of lg l in which each vertex other than v g l 1 is attached at most one pendant vertex a f g l graph is a f g l g r a p h in which vg is attached with pendant vertex denote by qmin the least q e i g e n v a l u e of a graph in this paper we proceed on considering the domination number the least q eigenvalue of a graph as well as their relation further results obtained are as follows i some results about the changing of the domination number under the structural perturbation of a graph are represented ii among all nonbipartite unicyclic graphs of order n with both domination number and girth g g n 1 the minimum qmin attains at a f g l graph for some l iii among the nonbipartite graphs of order n and with given domination number which contain a f g l graph as a subgraph some lower bounds for qmin are represented iv among the nonbipartite graphs of order n and with given domination number n 2

bounds on the q spread of a graph the spread s m of an n n complex matrix m is s m maxij i j where the maximum is taken over all pairs of eigenvalues of m i 1 i n 9 and 11 based on this concept gregory et al 7 determined some bounds for the spread of the adjacency matrix a g of a simple graph g and made a conjecture regarding the graph on n vertices yielding the maximum value of the spread of the corresponding adjacency matrix the signless laplacian matrix of a graph g q g d g a g where d g is the diagonal matrix of degrees of g and a g is its adjacency matrix has been recently studied 4 5 the main goal of this paper is to determine some bounds on s q g we prove that for any graph on n 5 vertices 2 s q g 2n 4 and we characterize the equality cases in both bounds further we prove that for any connected graph g with n 5 vertices s q g 2n 4 we conjecture that for n 5 sq g 4n 2 20n 33 and that in this case the upper bound is attained if and only if g is a certain pathcomplete graph

on the sign patterns of the smallest signless laplacian eigenvector abstract let h be a connected bipartite graph whose signless laplacian matrix is q h suppose that the bipartition of h is s t and that x is the eigenvector of the smallest eigenvalue of q h it is well known that x is positive and constant on s and negative and constant on t the resilience of the sign pattern of x under addition of edges into the subgraph induced by either s or t is investigated and a number of cases in which the sign pattern of x persists are described

a lower bound on the least signless laplacian eigenvalue of a graph let g be a simple graph with n vertices and m edges and let q n g be the least signless laplacian eigenvalue of g recently guo chen and yu proved that if n ge 6 then begin aligned q n g ge frac 2m n 2 n 1 end aligned which confirms a conjecture proposed by de lima oliveira de abreu and nikiforov in this note we present a new proof for the above inequality where we also completely characterize the case when the equality holds