Spectral Turán problem of non-bipartite graphs: Forbidden books
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2025
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| _version_ | 1866916781084377088 |
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| author | Liu, Ruifang Miao, Lu |
| author_facet | Liu, Ruifang Miao, Lu |
| contents | A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $ρ(G)\leqρ(T_{n,2})$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Turán problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For general $r\geq1$, let $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$ be the graph obtained from $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$ by adding a new vertex $v_{0}$ such that $v_{0}$ has exactly $r$ neighbours in each part of $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$. By adopting a different technique named the residual index, Chvátal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $ρ(G)\leqρ\Big(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}\Big)$ , with equality if and only if $G\cong K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$. An interesting phenomenon is that the spectral extremal graphs are completely different for $r=0$ and general $r\geq1$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_04884 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral Turán problem of non-bipartite graphs: Forbidden books Liu, Ruifang Miao, Lu Combinatorics 05C50, 05C35 A book graph $B_{r+1}$ is a set of $r+1$ triangles with a common edge, where $r\geq0$ is an integer. Zhai and Lin [J. Graph Theory 102 (2023) 502-520] proved that for $n\geq\frac{13}{2}r$, if $G$ is a $B_{r+1}$-free graph of order $n$, then $ρ(G)\leqρ(T_{n,2})$, with equality if and only if $G\cong T_{n,2}$. Note that the extremal graph $T_{n,2}$ is bipartite. Motivated by the above elegant result, we investigate the spectral Turán problem of non-bipartite $B_{r+1}$-free graphs of order $n$. For general $r\geq1$, let $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$ be the graph obtained from $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$ by adding a new vertex $v_{0}$ such that $v_{0}$ has exactly $r$ neighbours in each part of $K_{\lceil\frac{n-1}{2}\rceil,\lfloor\frac{n-1}{2}\rfloor}$. By adopting a different technique named the residual index, Chvátal-Hanson theorem and typical spectral extremal methods, we in this paper prove that: If $G$ is a non-bipartite $B_{r+1}$-free graph of order $n$, then $ρ(G)\leqρ\Big(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}\Big)$ , with equality if and only if $G\cong K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}^{r, r}$. An interesting phenomenon is that the spectral extremal graphs are completely different for $r=0$ and general $r\geq1$. |
| title | Spectral Turán problem of non-bipartite graphs: Forbidden books |
| topic | Combinatorics 05C50, 05C35 |
| url | https://arxiv.org/abs/2506.04884 |