The average connectivity matrix of a graph

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 $A_{\bar{\kappa }}\left(G\right)$, is an $n\times n$ matrix whose $(u,v)$-entry is $\kappa (u,v)/\left(\begin{array}{c}n\\ 2\end{array}\right)$ and let $\rho \left(A_{\bar{\kappa }}\right(G\left)\right)$ be the spectral radius of $A_{\bar{\kappa }}\left(G\right)$. 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 $\rho \left(A_{\bar{\kappa }}\right(G\left)\right)\le \frac{4{\alpha }^{\prime }\left(G\right)}{n}$, which implies a result of Kim and O [8] stating that for any connected graph G, we have $\bar{\kappa }\left(G\right)\le 2{\alpha }^{\prime }\left(G\right)$, where $\bar{\kappa }\left(G\right)={\sum }_{u,v\in V\left(G\right)}\frac{\kappa (u,v)}{\left(\begin{array}{c}n\\ 2\end{array}\right)}$ and ${\alpha }^{\prime }\left(G\right)$ 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 $\rho \left(A_{\bar{\kappa }}\right(G\left)\right)\le \frac{(n-{\alpha }^{\prime }(G\left)\right)\left(4{\alpha }^{\prime }\right(G)-2)}{n(n-1)}$, and equality in the bound holds only when G is a complete balanced bipartite graph.