On the sum of the first two largest signless Laplacian eigenvalues of a graph
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| Format: | Preprint |
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2023
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| _version_ | 1866911914754310144 |
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| author | Zhou, Zi-Ming He, Chang-Xiang Shan, Hai-Ying |
| author_facet | Zhou, Zi-Ming He, Chang-Xiang Shan, Hai-Ying |
| contents | For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2310_08880 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the sum of the first two largest signless Laplacian eigenvalues of a graph Zhou, Zi-Ming He, Chang-Xiang Shan, Hai-Ying Combinatorics For a graph $G$, let $S_2(G)$ be the sum of the first two largest signless Laplacian eigenvalues of $G$, and $f(G)=e(G)+3-S_2(G)$. Oliveira, Lima, Rama and Carvalho conjectured that $K^+_{1,n-1}$ (the star graph with an additional edge) is the unique graph with minimum value of $f(G)$ on $n$ vertices. In this paper, we prove this conjecture, which also confirm a conjecture for the upper bound of $S_2(G)$ proposed by Ashraf et al. |
| title | On the sum of the first two largest signless Laplacian eigenvalues of a graph |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.08880 |