Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs
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| Format: | Preprint |
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2024
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| _version_ | 1866914966444965888 |
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| author | Lew, Alan |
| author_facet | Lew, Alan |
| contents | Let $G=(V,E)$ be a graph on $n$ vertices, and let $λ_1(L(G))\ge \cdots\ge λ_{n-1}(L(G))\ge λ_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k λ_i(L(G)) \le |E|+\binom{k+1}{2}$. Here, we prove the following weak version of Brouwer's conjecture: For every $1\leq k \leq n$, \[
\sum_{i=1}^k λ_i(L(G)) \leq
|E|+k^2+15k\log{k}+65k. \] For a graph $G=(V,E)$, we define its partition density $\tildeρ(G)$ as the maximum, over all subgraphs $H$ of $G$, of the ratio between the number of edges of $H$ and the number of vertices in the largest connected component of $H$. Our argument relies on the study of the structure of the graphs $G$ satisfying $\tildeρ(G)< k$. In particular, using a result of Alon, McDiarmid and Reed, we show that every such graph can be decomposed into at most $k+ 15\log{k}+65$ edge-disjoint star forests (that is, forests whose connected components are all isomorphic to stars).
In addition, we show that for every graph $G=(V,E)$ and every $1\le k\le |V|$, \[
\sum_{i=1}^k λ_i(L(G)) \leq
|E|+k\cdot ν(G) + \left\lfloor\frac{k}{2}\right\rfloor, \] where $ν(G)$ is the maximum size of a matching in $G$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_04563 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs Lew, Alan Combinatorics Let $G=(V,E)$ be a graph on $n$ vertices, and let $λ_1(L(G))\ge \cdots\ge λ_{n-1}(L(G))\ge λ_n(L(G))=0$ be the eigenvalues of its Laplacian matrix $L(G)$. Brouwer conjectured that for every $1\le k\le n$, $\sum_{i=1}^k λ_i(L(G)) \le |E|+\binom{k+1}{2}$. Here, we prove the following weak version of Brouwer's conjecture: For every $1\leq k \leq n$, \[ \sum_{i=1}^k λ_i(L(G)) \leq |E|+k^2+15k\log{k}+65k. \] For a graph $G=(V,E)$, we define its partition density $\tildeρ(G)$ as the maximum, over all subgraphs $H$ of $G$, of the ratio between the number of edges of $H$ and the number of vertices in the largest connected component of $H$. Our argument relies on the study of the structure of the graphs $G$ satisfying $\tildeρ(G)< k$. In particular, using a result of Alon, McDiarmid and Reed, we show that every such graph can be decomposed into at most $k+ 15\log{k}+65$ edge-disjoint star forests (that is, forests whose connected components are all isomorphic to stars). In addition, we show that for every graph $G=(V,E)$ and every $1\le k\le |V|$, \[ \sum_{i=1}^k λ_i(L(G)) \leq |E|+k\cdot ν(G) + \left\lfloor\frac{k}{2}\right\rfloor, \] where $ν(G)$ is the maximum size of a matching in $G$. |
| title | Partition density, star arboricity, and sums of Laplacian eigenvalues of graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2410.04563 |