let $g$ be a simple graph of order $n$ and size $m$.the edge covering of $g$ is a set of edges such that every vertex of $g$ is incident to at least one edge of the set. the edge cover polynomial of $g$ is the polynomial$e(g,x)=sum_{i=rho(g)}^{m} e(g,i) x^{i}$,where $e(g,i)$ is the number of edge coverings of $g$ of size $i$, and$rho(g)$ is the edge covering number of $g$. in this paper we stud...

‎The excessive index of a bridgeless cubic graph $G$ is the least integer $k$‎, ‎such that $G$ can be covered by $k$ perfect matchings‎. ‎An equivalent form of Fulkerson conjecture (due to Berge) is that every bridgeless‎ ‎cubic graph has excessive index at most five‎. ‎Clearly‎, ‎Petersen graph is a cyclically 4-edge-connected snark with excessive index at least 5‎, ‎so Fouquet and Vanherpe as...

A graph   is called integral if all eigenvalues of its adjacency matrix  are integers.  Given a subset $S$ of a finite group $G$, the bi-Cayley graph $BCay(G,S)$ is a graph with vertex set $Gtimes{1,2}$ and edge set ${{(x,1),(sx,2)}mid sin S, xin G}$.  In this paper, we classify all finite groups admitting a connected cubic integral bi-Cayley graph.

‎A tree containing exactly two non-pendant vertices is called a double-star‎. ‎Let $k_1$ and $k_2$ be two positive integers‎. ‎The double-star with degree sequence $(k_1+1‎, ‎k_2+1‎, ‎1‎, ‎ldots‎, ‎1)$ is denoted by $S_{k_1‎, ‎k_2}$‎. ‎It is known that a cubic graph has an $S_{1,1}$-decomposition if and only if it contains a perfect matching‎. ‎In this paper‎, ‎we study the $S_{1,2}$-decomposit...

Let tsp(G) denote the length of a shortest travelling salesman tour in a graph G. We prove that for any ε > 0, there exists a simple 2-connected planar cubic graph G1 such that tsp(G1) ≥ (1.25 − ε) · |V (G1)|, a simple 2-connected bipartite cubic graph G2 such that tsp(G2) ≥ (1.2 − ε) · |V (G2)|, and a simple 3-connected cubic graph G3 such that tsp(G3) ≥ (1.125− ε) · |V (G3)|.