In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, two parameters Π1(G) and Π2(G), which are related coefficients of the characteristic polynomial graph G. All connected with Π1(G)∈{1,0,−1,−2,−3} Π2(G)∈{0,−1,−2,−3} characterized. Some interesting properties Π2(G) also given. We then find necessary sufficient conditions for classes be

We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs. use the Ihara-Bass formula to connect non-backtracking spectrum that adjacency matrix, employing moment method show there exists a matrix. A byproduct our main theorem is rectangular zero-one matrices with fixed row and column sums are full-rank high probability. Finally, we illustrate applications co...

In this paper, we define and obtain several properties of the (adjacency) energy a hypergraph. particular, bounds for are obtained as functions structural spectral parameters, such Zagreb index radius. We also study how hypergraph varies when vertex/edge is removed or an edge divided. addition, solve extremal problem class hyperstars, show that never odd number.

Eilers et al. have recently completed the geometric classification of unital graph $C^\ast$-algebras up to Morita equivalence using a set moves on corresponding digraphs. We explore question whether these preserve nonzero elements spectrum finite digraph, which in this paper is allowed loops and parallel edges. consider several different digraph spectra that been studied literature, answering f...

BACKGROUND Many complex systems can be represented as networks, and how a network breaks up into subnetworks or communities is of wide interest. However, the development of a method to detect nodes important to communities that is both fast and accurate is a very challenging and open problem. METHODOLOGY/PRINCIPAL FINDINGS In this manuscript, we introduce a new approach to characterize the no...

Recently, in Refs. [1] and [2], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes function associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel-Darboux identity. In this paper, evaluation of effec...

Let G = (V,E) be a graph without loops and multiple edges. Let n and m be the number of vertices and edges of G, respectively. Such a graph will be referred to as an (n,m)-graph. For v ∈ V (G), let d(v) be the degree of v. In this paper, we are concerned only with undirected simple graphs (loops and multiple edges are not allowed). Let G be a graph with n vertices and the adjacency matrix A(G)....

Abstract We establish the theory for pretty good state transfer in discrete-time quantum walks. For a class of walks, we show that is characterized by spectrum certain Hermitian adjacency matrix graph; more specifically, vertices involved must be strongly cospectral relative to this matrix, and arccosines its eigenvalues satisfy some number theoretic conditions. Using normalized matrices, cycli...

We study the average probability that a discrete-time quantum walk finds marked vertex on graph. first show that, for regular graph, spectrum of transition matrix is determined by weighted adjacency an augmented then consider search distance and find formula in terms its vertex-deleted subgraph. In particular, any family approaches 1/4 as valency goes to infinity. also present more relaxed crit...

A dumbbell graph, denoted by Da,b,c, is a bicyclic graph consisting of two vertexdisjoint cycles Ca, Cb and a path Pc+3 (c > −1) joining them having only its end-vertices in common with the two cycles. In this paper, we study the spectral characterization w.r.t. the adjacency spectrum of Da,b,0 (without cycles C4) with gcd(a, b) > 3, and we complete the research started in [J.F. Wang et al., A ...