Introduction As is well known, the combinatorial problem of counting paths of length n between two xed vertices in a graph reduces to raising the adjacency matrix A of the graph to the n-th power ((B], p. 11). For an undirected graph, A is symmetric and the problem above simpliies considerably if its spectrum (A) is known and contains few distinct elements. Spectra of graphs, meaning spectra of...

We compute the eigenvalue fluctuations of uniformly distributed random biregular bipartite graphs with fixed and growing degrees for a large class analytic functions. As key step in proof, we obtain total variation distance bound Poisson approximation number cycles cyclically non-backtracking walks graphs, which might be independent interest. also prove semicircle law [Formula: see text]-biregu...

Core-satellite graphs (sometimes referred to as generalized friendship graphs) are an interesting class of graphs that generalize many well known types of graphs. In this paper we show that two popular clustering measures, the average Watts-Strogatz clustering coefficient and the transitivity index, diverge when the graph size increases. We also show that these graphs are disassortative. In add...

A square (0, 1)-matrix X of order n ≥ 1 is called fully indecomposable if there exists no integer k with 1 ≤ k ≤ n− 1, such that X has a k by n− k zero submatrix. The reduced adjacency matrix of a bipartite graph G = (A,B,E) (having A ∪ B = {a1, ..., am} ∪ {b1, ..., bn} as vertex set, and E as edge set), is X = [xij ], 1 ≤ i ≤ m, 1 ≤ j ≤ n, where xij = 1 if aibj ∈ E and xij = 0 otherwise. A sta...

The spectrum of a matrix M is the multiset that contains all the eigenvalues of M. If M is a matrix obtained from a graph G, then the spectrum of M is also called the graph spectrum of G. If two graphs has the same spectrum, then they are cospectral (or isospectral) graphs. In this paper, we compare four spectra of matrices to examine their accuracy in protein structural comparison. These four ...

Let n be any positive integer and Fn be the friendship (or Dutch windmill) graph with 2n+1 vertices and 3n edges. Here we study graphs with the same adjacency spectrum as Fn. Two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. Let G be a graph cospectral with Fn. Here we prove that if G has no cycle of length 4 or 5, then G ∼= Fn. Moreover if G...

A matrix-weighted graph is an undirected with a k×k symmetric positive semidefinite matrix assigned to each edge. Such graphs admit natural generalizations of the Laplacian and adjacency matrices, leading generalized notion expansion. Extensions some theorems about expansion hold for graphs—in particular, analogue expander mixing lemma one half Cheeger-type inequality. These results lead defini...

We calculate the Smith normal form of the adjacency matrix of each of the following graphs or their complements (or both): complete graph, cycle graph, square of the cycle, power graph of the cycle, distance matrix graph of cycle, Andrásfai graph, Doob graph, cocktail party graph, crown graph, prism graph, Möbius ladder. The proofs operate by finding the abelianisation of a cyclically presented...

An oriented graph Gσ is a simple undirected graph G with an orientation, which assigns to each edge a direction so that Gσ becomes a directed graph. G is called the underlying graph of Gσ and we denote by S(Gσ) the skew-adjacency matrix of Gσ and its spectrum Sp(Gσ) is called the skew-spectrum of Gσ. In this paper, the coefficients of the characteristic polynomial of the skew-adjacency matrix S...