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- W4313305620 abstract "For a graph $G$ and for two distinct vertices $u$ and $v$, let $kappa(u,v)$ be the maximum number of vertex-disjoint paths joining $u$ and $v$ in $G$. The average connectivity matrix of an $n$-vertex connected graph $G$, written $A_{bar{kappa}}(G)$, is an $ntimes n$ matrix whose $(u,v)$-entry is $kappa(u,v)/{n choose 2}$ and let $rho(A_{bar{kappa}}(G))$ be the spectral radius of $A_{bar{kappa}}(G)$. In this paper, we investigate some spectral properties of the matrix. In particular, we prove that for any $n$-vertex connected graph $G$, we have $rho(A_{bar{kappa}}(G)) le frac{4alpha'(G)}n$, which implies a result of Kim and O cite{KO} stating that for any connected graph $G$, we have $bar{kappa}(G) le 2 alpha'(G)$, where $bar{kappa}(G)=sum_{u,v in V(G)}frac{kappa(u,v)}{{nchoose 2}}$ and $alpha'(G)$ is the maximum size of a matching in $G$; equality holds only when $G$ is a complete graph with an odd number of vertices. Also, for bipartite graphs, we improve the bound, namely $rho(A_{bar{kappa}}(G)) le frac{(n-alpha'(G))(4alpha'(G) - 2)}{n(n-1)}$, and equality in the bound holds only when $G$ is a complete balanced bipartite graph." @default.
- W4313305620 created "2023-01-06" @default.
- W4313305620 creator A5009202496 @default.
- W4313305620 creator A5055424683 @default.
- W4313305620 date "2022-12-28" @default.
- W4313305620 modified "2023-09-23" @default.
- W4313305620 title "The average connectivity matrix of a graph" @default.
- W4313305620 doi "https://doi.org/10.48550/arxiv.2212.13724" @default.
- W4313305620 hasPublicationYear "2022" @default.
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