arXiv subjects

Fengming Dong

Publications and source records attributed to Fengming Dong.

56 records · Page 4Linked to original sources

arXiv2017

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

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 $G$.

math.CO↗

arXiv2014

On Zero-free Intervals of Flow Polynomials

This article studies real roots of the flow polynomial $F(G,λ)$ of a bridgeless graph $G$. For any integer $k\ge 0$, let $ξ_k$ be the supremum in $(1,2]$ such that $F(G,λ)$ has no real roots in $(1,ξ_k)$ for all graphs $G$ with $|W(G)|\le k$, where $W(G)$ is the set of vertices in $G$ of degrees larger than $3$. We prove that $ξ_k$ can be determined by considering a finite set of graphs and show that $ξ_k=2$ for $k\le 2$, $ξ_3=1.430\cdots$, $ξ_4=1.361\cdots$ and $ξ_5=1.317\cdots$. We also prove that for any bridgeless graph $G=(V,E)$, if all roots of $F(G,λ)$ are real but some of these roots are not in the set $\{1,2,3\}$, then $|E|\ge |V|+17$ and $F(G,λ)$ has at least 9 real roots in $(1,2)$.

math.CO↗