Bollobás-Nikiforov Conjecture for graphs with not so many triangles
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866910544903012352 |
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| author | Kumar, Hitesh Pragada, Shivaramakrishna |
| author_facet | Kumar, Hitesh Pragada, Shivaramakrishna |
| contents | Bollobás and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ λ_1^2+λ_2^2\le \bigg( 1-\frac{1}{ω(G)}\bigg)2m\] where $λ_1$ and $λ_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $ω$ denotes the clique number of $G$. This conjecture was recently verified for triangle-free graphs by Lin, Ning and Wu and for regular graphs by Zhang. Elphick, Wocjan and Linz proposed a generalization of this conjecture. In this note, we verify this generalized conjecture for the family of graphs on $m$ edges, which contain at most $O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon > 0$. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bollobás-Nikiforov Conjecture for graphs with not so many triangles Kumar, Hitesh Pragada, Shivaramakrishna Combinatorics 05C50 Bollobás and Nikiforov conjectured that for any graph $G \neq K_n$ with $m$ edges \[ λ_1^2+λ_2^2\le \bigg( 1-\frac{1}{ω(G)}\bigg)2m\] where $λ_1$ and $λ_2$ denote the two largest eigenvalues of the adjacency matrix $A(G)$, and $ω$ denotes the clique number of $G$. This conjecture was recently verified for triangle-free graphs by Lin, Ning and Wu and for regular graphs by Zhang. Elphick, Wocjan and Linz proposed a generalization of this conjecture. In this note, we verify this generalized conjecture for the family of graphs on $m$ edges, which contain at most $O(m^{1.5-\varepsilon})$ triangles for some $\varepsilon > 0$. In particular, we show that the conjecture is true for planar graphs, book-free graphs and cycle-free graphs. |
| title | Bollobás-Nikiforov Conjecture for graphs with not so many triangles |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2407.19341 |