On the nullity of middle graphs

Let C be a connected graph, and let L(C) and M(C) be the line graph and middle graph of C. Gutman and Sciriha (On the nullity of line graphs of trees, Discrete Mathematics, 232 (2001), 35-45) proved that the nullity rl(L(T)) of L(T) of a tree T satisfies rl(L(T)) = 0 or rl(L(T)) = 1. But the problem to determine which trees T satisfy rl(L(T)) = 0 or rl(L(T)) = 1 is still open. In this paper, we prove that rl(M(C)) = 1 if C is a bipartite graph, and rl(M(C)) = 0 otherwise. As an application, we show that rl(C(n, m)) = 1 for the so-called silicate network C(n, m) obtained from the hexagonal lattice in the context of statistical physics. (c) 2025 Published by Elsevier Inc.