For \( u_1, u_2 \in V(G) \) , the closeness matrix of a graph \( G \) , denoted by \( C(G) \) , is a symmetric matrix where each entry is defined as \(c_G(u_1, u_2) = 2^{-d(u_1,u_2)}, \; \text {for } u_1 \ne u_2\) , and \(c_G(u_1, u_2) = 0, \; \text {if } u_1 = u_2\) , where \( d(u_1, u_2) \) represents the distance between vertices \( u_1 \) and \( u_2 \) in \( G \) . Let the eigenvalues of \( C(G) \) be ordered as \(\rho _1(C(G)) \ge \rho _2(C(G)) \ge \cdots \ge \rho _n(C(G))\) . In this paper, we identify graphs for which the second-largest closeness eigenvalue satisfies \( \rho _2(C(G)) \le 0 \) . Specifically, we characterize such graphs within the families of trees, unicyclic graphs, and \(K_4\) -minor-free multicyclic graphs.
