Expression for the Number of Spanning Trees of Line Graphs of Arbitrary Connected Graphs

TL;DR

This paper derives a formula for counting spanning trees in line graphs of subdivided graphs, generalizing previous results and confirming a conjecture for specific graph classes.

Contribution

It provides a new combinatorial expression for the number of spanning trees in line graphs of subdivided graphs, extending known relations and proving a conjecture.

Findings

Derived an explicit formula for t(L(S_r(G))) in terms of t(G)

Generalized known results on spanning trees of line graphs

Confirmed a conjecture for graphs with degree-1 vertices

Abstract

For any graph $G$, let $t(G)$ be the number of spanning trees of $G$, $L(G)$ be the line graph of $G$ and for any non-negative integer $r$, $S_r(G)$ be the graph obtained from $G$ by replacing each edge $e$ by a path of length $r+1$ connecting the two ends of $e$. In this paper we obtain an expression for $t(L(S_r(G)))$ in terms of spanning trees of $G$ by a combinatorial approach. This result generalizes some known results on the relation between $t(L(S_r(G)))$ and $t(G)$ and gives an explicit expression $t(L(S_r(G)))=k^{m+s-n-1}(rk+2)^{m-n+1}t(G)$ if $G$ is of order $n+s$ and size $m+s$ in which $s$ vertices are of degree $1$ and the others are of degree $k$. Thus we prove a conjecture on $t(L(S_1(G)))$ for such a graph $G$.