Strong reciprocal eigenvalue property of a class of weighted graphs

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• 作者：R.B. Bapat^a ; ¹ ; ^{rbb@isid.ac.in" class="auth_mail" title="E-mail the corresponding author} ; S.K. Panda^b ; ² ; ^{p.swarup@iitg.ernet.in" class="auth_mail" title="E-mail the corresponding author} ; S. Pati^b ; ^{pati@iitg.ernet.in" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C50 ; 15A18
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：511
• 期：Complete
• 页码：460-475
• 全文大小：419 K

文摘

Let H$\mathcal{H}$ be the class of connected bipartite graphs G   with a unique perfect matching M$\mathcal{M}$. For G∈H$G\in \mathcal{H}$, let W_G${\mathcal{W}}_G$ be the set of weight functions w on the edge set E(G)$E(G)$ such that w(e)=1$\mathrm{w}(e)=1$ for each matching edge and e406" title="Click to view the MathML source">w(e)>0$\mathrm{w}(e)>0$ for each nonmatching edge. Let G_w$G_{\mathrm{w}}$ denote the weighted graph with G∈H$G\in \mathcal{H}$ and w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$. The graph G_w$G_{\mathrm{w}}$ is said to satisfy the reciprocal eigenvalue property, property (R)  , if 1/λ$1/\lambda$ is an eigenvalue of the adjacency matrix A(G_w)$A(G_{\mathrm{w}})$ whenever λ   is an eigenvalue of A(G_w)$A(G_{\mathrm{w}})$. Moreover, if the multiplicities of the reciprocal eigenvalues are the same, we say G_w$G_{\mathrm{w}}$ has the strong reciprocal eigenvalue property, property (SR)  . Let H_g={G∈H|G/M is bipartite}${\mathcal{H}}_g=\{G\in \mathcal{H}|G/\mathcal{M}\text{is bipartite}\}$, where G/M$G/\mathcal{M}$ is the graph obtained from G   by contracting each edge in M$\mathcal{M}$ to a vertex.

Recently in [12], it was shown that if G∈H_g$G\in {\mathcal{H}}_g$, then G_w$G_{\mathrm{w}}$ has property (SR) for some w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$ if and only if G_w$G_{\mathrm{w}}$ has property (SR) for each w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$ if and only if G is a corona graph (obtained from another graph H by adding a new pendant vertex to each vertex of H).

Now we have the following questions. Is there a graph 18e78f03f3b3153d7" title="Click to view the MathML source">G∈H∖H_g$G\in \mathcal{H}\setminus {\mathcal{H}}_g$ such that G_w$G_{\mathrm{w}}$ has property (SR) for each w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$? Are there graphs 18e78f03f3b3153d7" title="Click to view the MathML source">G∈H∖H_g$G\in \mathcal{H}\setminus {\mathcal{H}}_g$ such that G_w$G_{\mathrm{w}}$ never has property (SR), not even for one w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$? Are there graphs G∈H$G\in \mathcal{H}$ such that G_w$G_{\mathrm{w}}$ has property (SR) for some w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$ but not for all w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$? In this article, we supply answers to these three questions. We also supply a graph class larger than H_g${\mathcal{H}}_g$ where for any graph G  , if G_w$G_{\mathrm{w}}$ has property (SR) for one w∈W_G$\mathrm{w}\in {\mathcal{W}}_G$, then G is a corona graph.