On the vertex-face graphs of triangulations

Abstract

Let $G=(V(G),E(G))$ be a triangulation with vertex set $V(G)=\{v_1,v_2, \ldots,v_n\}$ and edge set $E(G)$ embedded on an orientable surface with genus $g$. Define $G^{\nabla}$ to be the graph obtained from $G$ by inserting a new vertex $v_{\phi}$ to each face $\phi$ of $G$ and adding three new edges $(u,v_{\phi}),(v,v_{\phi})$ and $(w,v_{\phi})$, where $u, v$ and $w$ are the three vertices on the boundary of $\phi$. Let $G^{\Box}$ be the graph obtained from $G^{\nabla}$ by deleting all edges in $E(G)$ of $G^{\nabla}$. In this paper, first some spectral properties of $G^{\nabla}$ and $G^{\Box}$ are considered, then it is proved that $t(G^{\nabla})=3^{n+4g-3}5^{n-1}t(G)$ and $t(G^{\Box})=3^{n+4g-3}2^{n-1}t(G)$, where $t(G)$ is the number of spanning trees of $G$. As applications, the number of spanning trees and Kirchhoff indices of some lattices in the context of statistical physics are obtained.