arXiv subjects

Fengming Dong

Publications and source records attributed to Fengming Dong.

59 records · Page 4Linked to original sources

arXiv2018

From G-parking functions to B-parking functions

A matching $M$ in a multigraph $G=(V,E)$ is said to be uniquely restricted if $M$ is the only perfect matching in the subgraph of $G$ induced by $V(M)$ (i.e., the set of vertices saturated by $M$). For any fixed vertex $x_0$ in $G$, there is a bijection from the set of spanning trees of $G$ to the set of uniquely restricted matchings of size $|V|-1$ in $S(G)-x_0$, where $S(G)$ is the bipartite graph obtained from $G$ by subdividing each edge in $G$. Thus the notion "uniquely restricted matchings of a bipartite graph $H$ saturating all vertices in a partite set $X$" can be viewed as an extension of "spanning trees in a connected graph". Motivated by this observation, we extend the notion "G-parking functions" of a connected multigraph to "B-parking functions" $f:X\rightarrow \{-1,0,1,2,\cdots \}$ of a bipartite graph $H$ with a bipartition $(X,Y)$ and find a bijection $ψ$ from the set of uniquely restricted matchings of $H$ to the set of B-parking functions of $H$. We also show that for any uniquely restricted matching $M$ in $H$ with $|M|=|X|$, if $f=ψ(M)$, then $\sum_{x\in X}f(x)$ is exactly the number of elements $y\in Y-V(M)$ which are not externally B-active with respect to $M$ in $H$, where the new notion "externally B-active members with respect to $M$ in $H$" is an extension of "externally active edges with respect to a spanning tree in a connected multigraph".

math.CO↗

arXiv2018

On graphs whose flow polynomials have real roots only

Let $G$ be a bridgeless graph. In 2011 Kung and Royle showed that all roots of the flow polynomial $F(G,λ)$ of $G$ are integers if and only if $G$ is the dual of a chordal and plane graph. In this article, we study whether a bridgeless graph $G$ for which $F(G,λ)$ has real roots only must be the dual of some chordal and plane graph. We conclude that the answer of this problem for $G$ is positive if and only if $F(G,λ)$ does not have any real root in the interval $(1,2)$. We also prove that for any non-separable and $3$-edge connected $G$, if $G-e$ is also non-separable for each edge $e$ in $G$ and every $3$-edge-cut of $G$ consists of edges incident with some vertex of $G$, then all roots of $P(G,λ)$ are real if and only if either $G\in \{L,Z_3,K_4\}$ or $F(G,λ)$ contains at least $9$ real roots in the interval $(1,2)$, where $L$ is the graph with one vertex and one loop and $Z_3$ is the graph with two vertices and three parallel edges joining these two vertices.

math.CO↗

arXiv2017

Properties of chromatic polynomials of hypergraphs not held for chromatic polynomials of graphs

In this paper, we present some properties on chromatic polynomials of hypergraphs which do not hold for chromatic polynomials of graphs. We first show that chromatic polynomials of hypergraphs have all integers as their zeros and contain dense real zeros in the set of real numbers. We then prove that for any multigraph $G=(V,E)$, the number of totally cyclic orientations of $G$ is equal to the value of $|P(H,-1)|$, where $P(H,λ)$ is the chromatic polynomial of a hypergraph $H$ which is constructed from $G$. Finally we show that the multiplicity of root "$0$" of $P(H,λ)$ may be at least $2$ for some connected hypergraphs $H$, and the multiplicity of root "$1$" of $P(H,λ)$ may be $1$ for some connected and separable hypergraphs $H$ and may be $2$ for some connected and non-separable hypergraphs $H$.

math.CO↗

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 $(1,2)$.

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