In this paper we explore how a spectral technique suggested by quantum walks can be used to distinguish non-isomorphic cospectral graphs. Reviewing ideas from the field of quantum computing we recall the definition of the unitary matrices inducing quantum walks. We show how the spectra of these matrices are related to the spectra of the transition matrices of classical walks. Despite this relat...

Let a graph Γ be locally disjoint union of three copies of complete graphs Kq−1 and let Γ be cospectral with the Hamming graph H(3, q). Bang and Koolen [Asian-Eur. J. Math. 1 (2008), 147–156] proved that if q > 3, then Γ is isomorphic to H(3, q). We present a short proof of this result. AMS Mathematics Subject Classification (2000): 05C50, 05E30.

In this paper we will apply the tensor and its traces to investigate spectral characterization of unicyclic graphs. Let $G$ be a graph $G^m$ $m$-th power (hypergraph) $G$. The spectrum is referring adjacency matrix, tensor. called determined by high-ordered spectra (DHS for short) if, whenever $H$ such that $H^m$ cospectral with all $m$, then isomorphic first give formulas graphs, provide some ...

We consider a connected threshold graph G with A, S as its adjacency matrix and Seidel respectively. In this paper several spectral properties of are analysed. compute the characteristic polynomial determinant S. A formula for multiplicity eigenvalues $$\pm 1$$ characterisation graphs at most five distinct derived. Finally it is shown that two non isomorphic may be cospectral

A graph is said to be determined by the adjacency and Laplacian spectrum (or to be a DS graph, for short) if there is no other non-isomorphic graph with the same adjacency and Laplacian spectrum, respectively. It is known that connected graphs of index less than 2 are determined by their adjacency spectrum. In this paper, we focus on the problem of characterization of DS graphs of index less th...

For an incidence geometry $${\mathcal G}= ({\mathcal P}, {\mathcal L}, {\text {I}})$$ with a linear representation T}_2^*({\mathcal K})$$ , we apply WQH switching to construct non-geometric graph $$\Gamma '$$ cospectral the line $$ of G}$$ . As application, show that for $$h \ge 2$$ and $$0< m < h$$ there are strongly regular graphs parameters $$(v, k, \lambda \mu ) = (2^{2\,h} (2^{m+h}+2^m-2^h...

Two graphs are co-spectral if their respective adjacency matrices have the same multi-set of eigenvalues. A graph is said to be determined by its spectrum if all graphs that are co-spectral with it are isomorphic to it. We consider these properties in relation to logical definability. We show that any pair of graphs that are elementarily equivalent with respect to the three-variable counting fi...

Cámara and Haemers (2014) investigatedwhen a complete graphwith some edges deleted is determined by its adjacency spectrum (DAS for short). They claimed: for anym ≥ 6 and every large enough n one can obtain graphswhich are not DAS by removingm edges from a complete graph Kn. LetGn denote the set of all graphs obtained from a complete graph Kn by deleting six edges. In this paper, we show that a...