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Register a new userThe signless Laplacian spectral radius of graphs without intersecting odd cycles
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Let $F_{a_1,dots,a_k}$ be a graph consisting of $k$ cycles of odd length $2a_1+1,dots, 2a_k+1$, respectively which intersect in exactly a common vertex, where $kgeq1$ and $a_1ge a_2ge cdotsge a_kge 1$. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all $F_{a_1,dots,a_k}$-free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.
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Tur{a}n type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Tur{a}n type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for $kcdot P_3$ in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or $kcdot P_3$ as a subgraph, respectively.
It is well known that spectral Tur{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur{a}n type problem. Let $G$ be a graph and let $mathcal{G}$ be a set of graphs, we say $G$ is textit{$mathcal{G}$-free} if $G$ does not contain any element of $mathcal{G}$ as a subgraph. Denote by $lambda_1$ and $lambda_2$ the largest and the second largest eigenvalues of the adjacency matrix $A(G)$ of $G,$ respectively. In this paper we focus on the characterization of graphs without short odd cycles according to the adjacency eigenvalues of the graphs. Firstly, an upper bound on $lambda_1^{2k}+lambda_2^{2k}$ of $n$-vertex ${C_3,C_5,ldots,C_{2k+1}}$-free graphs is established, where $k$ is a positive integer. All the corresponding extremal graphs are identified. Secondly, a sufficient condition for non-bipartite graphs containing an odd cycle of length at most $2k+1$ in terms of its spectral radius is given. At last, we characterize the unique graph having the maximum spectral radius among the set of $n$-vertex non-bipartite graphs with odd girth at least $2k+3,$ which solves an open problem proposed by Lin, Ning and Wu [Eigenvalues and triangles in graphs, Combin. Probab. Comput. 30 (2) (2021) 258-270].
The odd wheel $W_{2k+1}$ is the graph formed by joining a vertex to a cycle of length $2k$. In this paper, we investigate the largest value of the spectral radius of the adjacency matrix of an $n$-vertex graph that does not contain $W_{2k+1}$. We determine the structure of the spectral extremal graphs for all $kgeq 2, k otin {4,5}$. When $k=2$, we show that these spectral extremal graphs are among the Tur{a}n-extremal graphs on $n$ vertices that do not contain $W_{2k+1}$ and have the maximum number of edges, but when $kgeq 9$, we show that the family of spectral extremal graphs and the family of Tur{a}n-extremal graphs are disjoint.
A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $ell_n^m$ with $n=2m$. In this paper, we characterize the case $ngeq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on $n$ vertices.
Let $S_{1}(m, d, k)$ be the $k$-uniform supertree obtained from a loose path $P:v_{1}, e_{1}, v_{2}, ldots,v_{d}, e_{d}, v_{d+1}$ with length $d$ by attaching $m-d$ edges at vertex $v_{lfloorfrac{d}{2}rfloor+1}.$ Let $mathbb{S}(m,d,k)$ be the set of $k$-uniform supertrees with $m$ edges and diameter $d$ and $q(G)$ be the signless Laplacian spectral radius of a $k$-uniform hypergraph $G$. In this paper, we mainly determine $S_{1}(m,d,k)$ with the largest signless Laplacian spectral radius among all supertrees in $mathbb{S}(m,d,k)$ for $3leq dleq m-1$. Furthermore, we determine the unique uniform supertree with the maximum signless Laplacian spectral radius among all the uniform supertrees with $n$ vertices and pendent edges (vertices).
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