Let 1 (G) : : : n (G) be the eigenvalues of the adjacency matrix of a graph G of order n; and G be the complement of G: Suppose F (G) is a …xed linear combination of i (G) ; n i+1 (G) ; i G ; and n i+1 G ; 1 i k: We show that the limit lim n!1 1 n max fF (G) : v (G) = ng always exists. Moreover, the statement remains true if the maximum is taken over some restricted families like “Kr-free”or “r...

The Laplacian of a directed graph G is the matrix L(G) = 0(G) — A(G), where A(G) is the adjacency matrix of G and O(G) the diagonal matrix of vertex outdegrees. The eigenvalues of G are the eigenvalues of A(G). Given a directed graph G we construct a derived directed graph D(G) whose vertices are the oriented spanning trees of G. Using a counting argument, we describe the eigenvalues of D(G) an...

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial–Meshulam model X(n, p) of random k-dimensional simplicial complexes on n vertices. We show that for p = Ω(log n/n), the eigenvalues of each of the matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of...

Let G be a graph with the adjacency matrix A(G), and let D(G) diagonal of degrees G. Nikiforov first defined Aα(G) as Aα(G)=αD(G)+(1−α)A(G), 0≤α≤1, which shed new light on A(G) Q(G)=D(G)+A(G), yielded some surprises. The α−adjacency energy EAα(G) is invariant that calculated from eigenvalues Aα(G). In this work, by combining theory structure properties, we provide upper lower bounds for in term...

This paper deals with adjacency matrices of signed cycle graphs and chemical descriptors based on them. The eigenvalues and eigenvectors of the matrices are calculated and their efficacy in classifying different signed cycles is determined. The efficacy of some numerical indices is also examined. Mathematics Subject Classification 2010: Primary 05C22; Secondary 05C50, 05C90

Extending to r > 1 a formula of the authors, we compute the expected reflection distance of a product of t random reflections in the complex reflection group G(r, 1, n). The result relies on an explicit decomposition of the reflection distance function into irreducible G(r, 1, n)-characters and on the eigenvalues of certain adjacency matrices.

Eigenvalue of a graph is the eigenvalue of its adjacency matrix. The energy of a graph is the sum of the absolute values of its eigenvalues. In this note we obtain analytic expressions for the energy of two classes of regular graphs.

Different classes of communication network topologies and their representation in the form of adjacency matrix and its eigenvalues are presented. A self-organizing feature map neural network is used to map different classes of communication network topological patterns. The neural network simulation results are reported.

We refute, improve or amplify some recent results on graph eigenvalues. In particular, we prove that if G is a graph of order n ≥ 2, maximum degree ∆, and girth at least 5, then the maximum eigenvalue μ (G) of the adjacency matrix of G satisfies μ (G) ≤ min {

A multilevel circulant is defined as a graph whose adjacency matrix has a certain block decomposition into circulant matrices. A general algebraic method for finding the eigenvectors and the eigenvalues of multilevel circulants is given. Several classes of graphs, including regular polyhedra, suns, and cylinders can be analyzed using this scheme.