Generalized Spectral Characterizations of General Graphs Based on Graph-vectors
Given a graph G ,if for any graph H having the same spectrum with G and their complement graphs also sharing the same spectrumthe with the complement graph of G is isomorphic to G ,then the graph G is said to be determined by its generalized spectrum.In [12],the authors present a simple arith-metic condition to characterize if a graph is determined by its generalized spectrum.Specifically,for a graph G with n vertices,let its adjacency matrix beA=A(G)and its walk matrix beW(G)=[e,Ae,…, An-1e](where e is an all-one vector).If 2-[n/2]detW(G)is odd and square-free,then the graph G is deter-mined by its generalized spectrum.In [7],the authors introduce the concept of graph-vectors and provide a new matrix related to the graph G ,extend the concept about characterizing graphs by their spectra to Φ-DS ,and prove a class of regular graphs is Φ-DS .Building upon [7],this paper proposes a simple cri-terion to characterize a class of general graphs to be Φ-DS ,extends the conclusions mentioned in [12], and validates the effectiveness of the criterion through numerical experiments.
graph spectrumgeneralized spectrumgraph-vectorsDSSmith Normal Form
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维普
