Let G be a Kr+1-free graph with n vertices and m edges, and let n (G) be the smallest eigenvalue of its adjacency matrix. We show that

The characterization of a graph via the variable adjacency matrix enables to de ne a partially ordered relation on the walks. Studying the incidence algebra on this poset reveals unsuspected relations between connected and self-avoiding walks on the graph. These relations are derived by considering truncated versions of the characteristic polynomial of variable adjacency matrix, resulting in a ...

The energy of a molecular graph G is defined as the summation of the absolute values of the eigenvalues of adjacency matrix of a graph G. In this paper, an infinite class of fullerene graphs with 10n vertices, n ≥ 2, is considered. By proving centrosymmetricity of the adjacency matrix of these fullerene graphs, a lower bound for its energy is given. Our method is general and can be extended to ...

Let G be a graph with n vertices and m edges and let μ1 (G) ≥ ... ≥ μn (G) be the eigenvalues of its adjacency matrix. We discuss the following general problem. For k fixed and n large, find or estimate fk (n) = max v(G)=n |μk (G)|+ ∣

Suppose k is a positive integer, G and H are graphs, and / is a k-to-l correspondence from a vertex set of G onto a vertex set of H. Conditions on the adjacency matrices are given that are necessary and sufficient for / to extend to a continuous k-to-l map from G onto H .

The rank of a graph is that of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. We determine the maximum order of reduced trees as well as bipartite graphs with a given rank and characterize those graphs achieving the maximum order.

The (n, k)-arrangement graph A(n, k) is a graph with all the k-permutations of an n-element set as vertices where two k-permutations are adjacent if they agree in exactly k − 1 positions. We introduce a cyclic decomposition for k-permutations and show that this gives rise to a very fine equitable partition of A(n, k). This equitable partition can be employed to compute the complete set of eigen...

Let r ≥ 2 and c > 0. If G is a graph of order n and the largest eigenvalue of its adjacency matrix satisfies μ (G) ≥ (1− 1/r + c)n, then G contains a complete r-partite subgraph with r − 1 parts of size ⌊

The notions of subgraph centrality and communicability, based on the exponential of the adjacency matrix of the underlying graph, have been effectively used in the analysis of undirected networks. In this paper we propose an extension of these measures to directed networks, and we apply them to the problem of ranking hubs and authorities. The extension is achieved by bipartization, i.e., the di...