The closeness eigenvalues of graphs

Abstract

For a graph G with $u,v\in V(G)$, denote by $d_G(u,v)$ the distance between u and v in G, which is the length of a shortest path connecting them if there is at least one path from u to v in G and is $\infty$ otherwise. The closeness matrix of a graph G is the $|V(G)|\times |V(G)|$ symmetric matrix $(c_G(u,v))_{u,v\in V(G}$, where $c_G(u,v)=2^{-d_G(u,v)}$ if $u\ne v$ and 0 otherwise. The closeness eigenvalues of a graph G are the eigenvalues of C(G). We determine the graphs for which the second largest closeness eigenvalues belong to $\left( -\infty , a\right)$, where $a\approx -0.1571$ is the second largest root of $x^3-x^2-\frac{11}{8}x-\frac{3}{16}=0$. We also identify the n-vertex graphs with a closeness eigenvalue of multiplicity $n-1$, $n-2$ and $n-3$, respectively.