More results on the spectral radius of graphs with no odd wheels
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910559008456704 |
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| author | Zhang, Wenqian |
| author_facet | Zhang, Wenqian |
| contents | For a graph $G$, the spectral radius $λ_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all the vertices of the cycle. Let ${\rm SPEX}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order $n$ with the maximum spectral radius. Very recently, Cioabă, Desai and Tait \cite{CDT2} characterized the graphs in ${\rm SPEX}(n,W_{2k+1})$ for sufficiently large $n$, where $k\geq2$ and $k\neq4,5$. And they left the case $k=4,5$ as a problem. In this paper, we settle this problem. Moreover, we completely characterize the graphs in ${\rm SPEX}(n,W_{2k+1})$ when $k\geq4$ is even and $n\equiv2~(\mod4)$ is sufficiently large. Consequently, the graphs in ${\rm SPEX}(n,W_{2k+1})$ are characterized completely for any $k\geq2$ and sufficiently large $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_03595 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | More results on the spectral radius of graphs with no odd wheels Zhang, Wenqian Combinatorics 05C50 For a graph $G$, the spectral radius $λ_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all the vertices of the cycle. Let ${\rm SPEX}(n,W_{2k+1})$ be the set of $W_{2k+1}$-free graphs of order $n$ with the maximum spectral radius. Very recently, Cioabă, Desai and Tait \cite{CDT2} characterized the graphs in ${\rm SPEX}(n,W_{2k+1})$ for sufficiently large $n$, where $k\geq2$ and $k\neq4,5$. And they left the case $k=4,5$ as a problem. In this paper, we settle this problem. Moreover, we completely characterize the graphs in ${\rm SPEX}(n,W_{2k+1})$ when $k\geq4$ is even and $n\equiv2~(\mod4)$ is sufficiently large. Consequently, the graphs in ${\rm SPEX}(n,W_{2k+1})$ are characterized completely for any $k\geq2$ and sufficiently large $n$. |
| title | More results on the spectral radius of graphs with no odd wheels |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2408.03595 |