On the nullity of middle graphs

Abstract

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