Eigenvalues and triangles in graphs
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866908595655802880 |
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| author | Lin, Huiqiu Ning, Bo Wu, Baoyindureng |
| author_facet | Lin, Huiqiu Ning, Bo Wu, Baoyindureng |
| contents | Bollobás and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $λ^2_1(G)+λ^2_2(G)\leq \frac{r-1}{r}\cdot2m$, where $λ_1(G)$ and $λ_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $λ_1(G)\geq\sqrt{m-1}$ and $G\neq C_5\cup (n-5)K_1$; and (2) $λ_1(G)\geq λ_1(S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}))$ and $G\neq S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$, where $S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$ is obtained from $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_12474 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Eigenvalues and triangles in graphs Lin, Huiqiu Ning, Bo Wu, Baoyindureng Combinatorics 05C50 Bollobás and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $λ^2_1(G)+λ^2_2(G)\leq \frac{r-1}{r}\cdot2m$, where $λ_1(G)$ and $λ_2(G)$ are the largest and the second largest eigenvalues of the adjacency matrix $A(G)$, respectively. In this paper, we confirm the conjecture in the case $r=2$, by using tools from doubly stochastic matrix theory, and also characterize all families of extremal graphs. Motivated by classic theorems due to Erdős and Nosal respectively, we prove that every non-bipartite graph $G$ of order $n$ and size $m$ contains a triangle, if one of the following is true: (1) $λ_1(G)\geq\sqrt{m-1}$ and $G\neq C_5\cup (n-5)K_1$; and (2) $λ_1(G)\geq λ_1(S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}))$ and $G\neq S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$, where $S(K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil})$ is obtained from $K_{\lfloor\frac{n-1}{2}\rfloor,\lceil\frac{n-1}{2}\rceil}$ by subdividing an edge. Both conditions are best possible. We conclude this paper with some open problems. |
| title | Eigenvalues and triangles in graphs |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/1910.12474 |