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- W2894815576 abstract "Let $G$ be a (multi)graph of order $n$ and let $u,v$ be vertices of $G$. The maximum number of internally disjoint $u$-$v$ paths in $G$ is denoted by $kappa_G(u,v)$, and the maximum number of edge-disjoint $u$-$v$ paths in $G$ is denoted by $lambda_G (u,v)$. The average connectivity of $G$ is defined by $overline{kappa}(G)=sum_{{u,v}subseteq V(G)} kappa_G(u,v)/tbinom{n}{2},$ and the average edge-connectivity of $G$ is defined by $overline{lambda}(G)=sum_{{u,v}subseteq V(G)} lambda_G(u,v)/tbinom{n}{2}$. A graph $G$ is called ideally connected if $kappa_G(u,v)=min{mathrm{deg}(u),mathrm{deg}(v)}$ for all pairs of vertices ${u,v}$ of $G$. We prove that every minimally $2$-connected graph of order $n$ with largest average connectivity is bipartite, with the set of vertices of degree $2$ and the set of vertices of degree at least $3$ being the partite sets. We use this structure to prove that $overline{kappa}(G)<tfrac{9}{4}$ for any minimally $2$-connected graph $G$. This bound is asymptotically tight, and we prove that every extremal graph of order $n$ is obtained from some ideally connected nearly regular graph on roughly $n/4$ vertices and $3n/4$ edges by subdividing every edge. We also prove that $overline{lambda}(G)<tfrac{9}{4}$ for any minimally $2$-edge-connected graph $G$, and provide a similar characterization of the extremal graphs." @default.
- W2894815576 created "2018-10-12" @default.
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- W2894815576 date "2018-10-03" @default.
- W2894815576 modified "2023-10-01" @default.
- W2894815576 title "Average connectivity of minimally 2-connected graphs and average edge-connectivity of minimally 2-edge-connected graphs" @default.
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