Maximizing the signless Laplacian spectral radius of some theta graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866913607910948864 |
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| author | Liu, Yuxiang Wang, Ligong |
| author_facet | Liu, Yuxiang Wang, Ligong |
| contents | Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. Let $θ(l_{1},l_{2},l_{3})$ denote the theta graph which consists of two vertices connected by three internally disjoint paths with length $l_{1}$, $l_{2}$ and $l_{3}$. Let $F_{n}$ be the friendship graph consisting of $\frac{n-1}{2}$ triangles which intersect in exactly one common vertex for odd $n\geq3$ and obtained by hanging an edge to the center of $F_{n-1}$ for even $n\geq4$. Let $S_{n,k}$ denote the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices. Let $S_{n,k}^{+}$ denote the graph obtained by adding an edge to the two isolated vertices of $S_{n,k}$. In this paper, firstly, we show that if $G$ is $θ(1,2,2)$-free, then $q(G)\leq q(F_{n})$, unless $G\cong F_{n}$. Secondly, we show that if $G$ is $θ(1,2,3)$-free, then $q(G)\leq q(S_{n,2})$, unless $G\cong S_{n,2}$. Finally, we show that if $G$ is $\{θ(1,2,2),F_{5}\}$-free, then $q(G)\leq q(S_{n,1}^{+})$, unless $G\cong S_{n,1}^{+}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_08417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Maximizing the signless Laplacian spectral radius of some theta graphs Liu, Yuxiang Wang, Ligong Combinatorics Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph $G$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. Let $θ(l_{1},l_{2},l_{3})$ denote the theta graph which consists of two vertices connected by three internally disjoint paths with length $l_{1}$, $l_{2}$ and $l_{3}$. Let $F_{n}$ be the friendship graph consisting of $\frac{n-1}{2}$ triangles which intersect in exactly one common vertex for odd $n\geq3$ and obtained by hanging an edge to the center of $F_{n-1}$ for even $n\geq4$. Let $S_{n,k}$ denote the graph obtained by joining each vertex of $K_{k}$ to $n-k$ isolated vertices. Let $S_{n,k}^{+}$ denote the graph obtained by adding an edge to the two isolated vertices of $S_{n,k}$. In this paper, firstly, we show that if $G$ is $θ(1,2,2)$-free, then $q(G)\leq q(F_{n})$, unless $G\cong F_{n}$. Secondly, we show that if $G$ is $θ(1,2,3)$-free, then $q(G)\leq q(S_{n,2})$, unless $G\cong S_{n,2}$. Finally, we show that if $G$ is $\{θ(1,2,2),F_{5}\}$-free, then $q(G)\leq q(S_{n,1}^{+})$, unless $G\cong S_{n,1}^{+}$. |
| title | Maximizing the signless Laplacian spectral radius of some theta graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2412.08417 |