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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si1.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=2be55cdbb5ee9718d33c6234249e6888" title="Click to view the MathML source">a(k)=(a₁,a₂,…,a_k)class="mathContainer hidden">class="mathCode">$\mathbf{a}(\mathbf{k})=(a_1,a_2,\dots ,a_k)$ be a sequence of positive integers. A butterfly-like graph  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si121.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=5f2dba172b36c82f74f2d467b1c8f852" title="Click to view the MathML source">W_p^(s);a(k)class="mathContainer hidden">class="mathCode">$W_{p^{(s)};\mathbf{a}(\mathbf{k})}$ is a graph consisting of s  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si3.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=a0b4d7bd2bab5ced562d0d242091046e" title="Click to view the MathML source">(≥1)class="mathContainer hidden">class="mathCode">$(\ge 1)$ cycle of lengths class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si4.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=55be786f5897a1a164de5b1d936dbdcb" title="Click to view the MathML source">p+1class="mathContainer hidden">class="mathCode">$p+1$, and k  class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si3.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=a0b4d7bd2bab5ced562d0d242091046e" title="Click to view the MathML source">(≥1)class="mathContainer hidden">class="mathCode">$(\ge 1)$ paths class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si5.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=3b6d2de856f3932200bf00232d4fc73f" title="Click to view the MathML source">P_a1+1class="mathContainer hidden">class="mathCode">$P_{a_1+1}$, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si6.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=d13fb0e4451ece43247b9166206eacb3" title="Click to view the MathML source">P_a2+1class="mathContainer hidden">class="mathCode">$P_{a_2+1}$, …, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si7.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=b2bf07de72f0f0f850c565e0c99fc6d5" title="Click to view the MathML source">P_ak+1class="mathContainer hidden">class="mathCode">$P_{a_k+1}$ 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si289.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=c5c028a6685a064796dca3aaccf0e6e0" title="Click to view the MathML source">L−DSclass="mathContainer hidden">class="mathCode">$L-DS$ (respectively, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si288.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=650d83b413c9054009cbeab84ed0c51c" title="Click to view the MathML source">Q−DSclass="mathContainer hidden">class="mathCode">$Q-DS$). 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si10.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=e70688745a22904a0fec00e028f806be" title="Click to view the MathML source">L(G)class="mathContainer hidden">class="mathCode">$L(G)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si11.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=454e1569a1e2ff8ae32ee7563cd97472" title="Click to view the MathML source">Q(G)class="mathContainer hidden">class="mathCode">$Q(G)$, 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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si12.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=8f2be4db2a5447060a7242f531c7525f" title="Click to view the MathML source">W_2^(s);a(k)class="mathContainer hidden">class="mathCode">$W_{2^{(s)};\mathbf{a}(\mathbf{k})}$ are both class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si288.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=650d83b413c9054009cbeab84ed0c51c" title="Click to view the MathML source">Q−DSclass="mathContainer hidden">class="mathCode">$Q-DS$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0024379516304633&_mathId=si289.gif&_user=111111111&_pii=S0024379516304633&_rdoc=1&_issn=00243795&md5=c5c028a6685a064796dca3aaccf0e6e0" title="Click to view the MathML source">L−DSclass="mathContainer hidden">class="mathCode">$L-DS$.