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...

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...

Let G be a simple graph of order n and size m. An edge covering of the graph G is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Let e(G, k) be the number of edge covering sets of G of size k. The edge cover polynomial of G is the polynomial E(G, x) = m

Let G be a simple graph of order n and size m. An edge covering of a graph is a set of edges such that every vertex of the graph is incident to at least one edge of the set. Here we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G, x) = ∑m i=1 e(G, i)x , where e(G, i) is the number of edge covering sets of G of size i. Let G and H be two graphs of order n s...

abstract. suppose g is an nvertex and medge simple graph with edge set e(g). an integervalued function f: e(g) → z is called a flow. tutte was introduced the flow polynomial f(g, λ) as a polynomial in an indeterminate λ with integer coefficients by f(g,λ) in this paper the flow polynomial of some dendrimers are computed.

The combinatorial Alexander dual of the independence complex Ind(G) and that of the edge covering complex EC(G) are shown to have isomorphic homology groups for each non-null graph G. This yields isomorphisms of homology groups of Ind(G) and EC(G) with homology dimensions being appropriately shifted and restricted. The results exhibits the complementary nature of homology groups of Ind(G) and E...

In this paper we study problems where one has to prove that certain graph parameter attains its maximum at the star and its minimum at the path among the trees on a fixed number of vertices. We give many applications of the socalled generalized tree shift which seems to be a powerful tool to attack the problems of the above mentioned kind. We show that the generalized tree shift increases the l...

An edge cover of a graph is a set of edges in which each vertex has at least one of its incident edges. The problem of counting the number of edge covers is #P-complete and was shown to admit a fully polynomial-time approximation scheme (FPTAS) recently [10]. Counting weighted edge covers is the problem of computing the sum of the weights for all the edge covers, where the weight of each edge c...

Suppose G is an nvertex and medge simple graph with edge set E(G). An integervalued function f: E(G) → Z is called a flow. Tutte was introduced the flow polynomial F(G, λ) as a polynomial in an indeterminate λ with integer coefficients by F(G,λ) In this paper the Flow polynomial of some dendrimers are computed.