The spectral characterization of butterfly-like graphs

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• 作者：Muhuo Liu^a ; ^{liumuhuo@163.com" class="auth_mail" title="E-mail the corresponding author} ; Yanli Zhu^a ; ^{yanlizhu130@126.com" class="auth_mail" title="E-mail the corresponding author} ; Haiying Shan^b ; ^{shan_haiying@tongji.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Kinkar Ch. Das^c ; ^{kinkardas2003@googlemail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：05C50 ; 15A18 ; 15A36
• 刊名：Linear Algebra and its Applications
• 出版年：2017
• 出版时间：15 January 2017
• 年：2017
• 卷：513
• 期：Complete
• 页码：55-68
• 全文大小：393 K

文摘

Let a(k)=(a₁,a₂,…,a_k)$\mathbf{a}(\mathbf{k})=(a_1,a_2,\dots ,a_k)$ be a sequence of positive integers. A butterfly-like graph  W_p^(s);a(k)$W_{p^{(s)};\mathbf{a}(\mathbf{k})}$ is a graph consisting of s  (≥1)$(\ge 1)$ cycle of lengths p+1$p+1$, and k  (≥1)$(\ge 1)$ paths P_a1+1$P_{a_1+1}$, P_a2+1$P_{a_2+1}$, …, P_ak+1$P_{a_k+1}$ intersecting in a single vertex. The girth of a graph G is the length of a shortest cycle in G. Two graphs are said to be A-cospectral if they have the same adjacency spectrum. For a graph G, if there does not exist another non-isomorphic graph H such that G and H share the same Laplacian (respectively, signless Laplacian) spectrum, then we say that G   is L−DS$L-DS$ (respectively, Q−DS$Q-DS$). In this paper, we firstly prove that no two non-isomorphic butterfly-like graphs with the same girth are A-cospectral, and then present a new upper and lower bounds for the i  -th largest eigenvalue of L(G)$L(G)$ and Q(G)$Q(G)$, respectively. By applying these new results, we give a positive answer to an open problem in Wen et al. (2015) [17] by proving that all the butterfly-like graphs W_2^(s);a(k)$W_{2^{(s)};\mathbf{a}(\mathbf{k})}$ are both Q−DS$Q-DS$ and L−DS$L-DS$.