Closeness spectra and structural uniqueness of special graph classes

Abstract

For a graph G with vertex set $V\left(G\right)$ and edge set $E\left(G\right)$. Let $d(u,v)$ be the distance between vertices u and v. The closeness matrix of a graph G is a symmetric matrix, where each entry $c_G(u,v)$ is defined as $c_G(u,v)=2^{-d(u,v)}$ for $u\ne v$, and $c_G(u,v)=0$, if $u=v$. In the present study, we investigate the closeness spectra of connected graphs and inquire whether specific graph classes may be distinguished by their closeness eigenvalues. We inspect the tree $T_{3,n-3}$, the path $P_n$, the complete graph $K_n$, and the join graph $(P_1\cup P_3)\vee \overline{K_{n-4}}$. Applying careful spectral computation, we demonstrate that such graphs are uniquely specified by their closeness spectra. Additionally, we confirm that the tree $T_{3,n-3}$ achieves the second-smallest closeness eigenvalue among trees of diameter 3. Our findings emphasize the importance of closeness matrices in spectral graph theory and lead to more clarity of the relationship between the structure of graphs and its spectrum.