Given a digraph D, the complementarity spectrum of is defined as set eigenvalues its adjacency matrix. This has been shown to be useful in several fields, particularly spectral graph theory. The differences between properties for (undirected) graphs and digraphs, makes study latter particular interest, characterizing strongly connected digraphs with small number non trivial problem. Recently, o...

We give an elementary combinatorial proof of Bass’s determinant formula for the zeta function of a finite regular graph. This is done by expressing the number of non-backtracking cycles of a given length in terms of Chebychev polynomials in the eigenvalues of the adjacency operator of the graph.

A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees. It is also shown that integral trees with nullity 2 and 3 are unique.

We derive a new upper bound for the diameter of a k-regular graphG as a function of the eigenvalues of the adjacency matrix. Namely, supposethe adjacency matrix of G has eigenvalues AI , A2 .••.• An with lAd:::: IA21 ::::... :::: IAnl where AI = k, A = IA21. Then the diameter D(G) must satisfy D(G) :::; rlog(n 1)f1og(k/A)l. We wilJ consider families of graphs whose eigenvalues can b...

The distance energy of a graph G is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of G. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the ...

In graph signal processing, the graph adjacency matrix or the graph Laplacian commonly define the shift operator. The spectral decomposition of the shift operator plays an important role in that the eigenvalues represent frequencies and the eigenvectors provide a spectral basis. This is useful, for example, in the design of filters. However, the graph or network may be uncertain due to stochast...

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 ...

We consider a bipartite distance-regular graph Γ with diameter D ≥ 4 and valency k ≥ 3. Let X denote the vertex set of Γ and fix x ∈ X. Let Γ22 denote the graph with vertex set X̆ = {y ∈ X | ∂(x, y) = 2}, and edge set R̆ = {yz | y, z ∈ X̆, ∂(y, z) = 2}, where ∂ is the path-length distance function for Γ. The graph Γ22 has exactly k2 vertices, where k2 is the second valency of Γ. Let η1, η2, . . . ...

The purpose of this article is to improve existing lower bounds on the chromatic number χ. Let μ1, . . . , μn be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound χ > 1 + maxm{ ∑m i=1 μi/ − ∑m i=1 μn−i+1} for m = 1, . . . , n − 1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case...

A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly 27 connected integral Cayley graphs up to 11 vertices.