A remark on the generalized spectral characterization of the disjoint union of graphs

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• 作者：Wei Wang ; ^{wang_weiw@163.com} ; Lihuan Mao
• 关键词：05C50
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：1 April 2017
• 年：2017
• 卷：518
• 期：Complete
• 页码：1-13
• 全文大小：324 K
• 卷排序：518

文摘

A graph G is said to be determined by its generalized spectrum (DGS for short) if whenever Γ is a graph such that Γ and G are cospectral with cospectral complements, then Γ is isomorphic to G  . Let G∪HG∪H be the disjoint union of graphs G and H  . In this paper, we give a simple sufficient condition, under which we show that G∪HG∪H is DGS if and only if both G and H   are DGS. In particular, let H={x}H={x} be a singleton graph, we show that if gcd⁡(an,det⁡(W(G)))=1gcd⁡(an,det⁡(W(G)))=1 and anan is square-free, then G∪{x}G∪{x} is DGS if and only if G   is DGS, where anan is the constant term of the characteristic polynomial of G   and W(G)W(G) is the walk-matrix of G. It is noticed that in Wang and Xu [9], the authors gave a sufficient condition for G∪{x}G∪{x} to be DGS if G   is DGS. However, they missed the condition that anan is square-free in their theorem, and the result obtained is incorrect. We found a counterexample to their result without this condition and give a correct version of the result accordingly in this paper.