A (k,g)-cage is a k-regular simple graph of girth g with minimum possible number vertices. In this paper, (k,g)-cages which are Moore graphs referred as minimal (k,g)-cages. connected called distance regular (DR) if all its vertices have the same intersection array. bipartite biregular (DBR) partite set admit It known that DR and their subdivisions DBR graphs. for we give formula spectral radiu...

It is well known that if all the elementary symmetric functions of the eigenvalues of an n–by–n matrix are positive, then all its eigenvalues lie in the region of the complex plane {z : −π + πn < argz < π − π n}. Let A denote an n–by–n matrix with all diagonal entries nonzero and for which the length of the longest cycle in its directed graph is k, 2 ≤ k ≤ n. If, in addition, all the cycles in ...

Let G be a simple graph with adjacency matrix A (= AG). The eigenvalues and the spectrum of A are also called the eigenvalues and the spectrum of G, respectively. If we consider a matrix Q = D + A instead of A, where D is the diagonal matrix of vertex–degrees (in G), we get the signless Laplacian eigenvalues and the signless Laplacian spectrum, respectively. For short, the signless Laplacian ei...

If G is a simple graph (a non-oriented graph without loops or multiple edges), its (0, 1)-adjacency matrix A is symmetric and roots of the characteristic polynomial PG (λ) = det (λI −A) (the eigenvalues of G, making up its spectrum) are all real numbers, for which we assume their non-increasing order: λ1 ≥ λ2 ≥ · · · ≥ λn. In a connected graph for the largest eigenvalue λ1 (the index of the gra...

We study a generalization of strongly regular graphs. We call a graph strongly walkregular if there is an ` > 1 such that the number of walks of length ` from a vertex to another vertex depends only on whether the two vertices are the same, adjacent, or not adjacent. We will show that a strongly walk-regular graph must be an empty graph, a complete graph, a strongly regular graph, a disjoint un...

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We consider the problem of achieving average consensus in the minimum number of linear iterations on a fixed, undirected graph. We are motivated by the task of deriving lower bounds for consensus protocols and by the so-called “definitive consensus conjecture” which states that for an undirected connected graph G with diameter D there exist D matrices whose nonzero-pattern complies with the edg...

There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière used Riemannian metrics concentrated in a small neighborhood of a graph to prove that for every manifold M of dimension greater ...