On coefficients of edge domination polynomial of a graph

Abstract

An edge domination polynomial of a graph G is the polynomialis the number of edge dominating sets of G of cardinality t. In this paper, we provide tables which contain coefficient of edge domination polynomial of path and cycle. Also, certain properties of edge dominating polynomial are given.

Pendant Total Domination Polynomial of Some Families of Standard Graphs

“…Cockayne et al [5] introduced the definition of total domination, and later in 2014 Chaluvaraju and Chaitra [4] defined the total domination polynomial which is given as: "A total domination polynomial of graph G of order n is the polynomial D td (G, x) = n t=γ td (G) d td (G, t)x t , where d td (G, t) is the number of total dominating sets of graph G of cardinality t".…”

Some applications of Wagner's weighted subgraph counting polynomial

“…Similarly, for a graph G a set S ⊆ V (G) is called a total dominating set if for every u ∈ V (G) there exists a neighbor v ∈ S of u. Let d G,t (k) denote the number of dominating sets of size k. The domination polynomial is defined as [9,12] for further details.…”

Total domination polynomials of graphs

“…and edge set E(G 1 ) ∪ E(G 2 ), G 1 ∨ G 2 is called the join graph of G 1 and G 2 with vertex set V (G 1 ) ∪ V (G 2 ) and edge set E(G 1 ) ∪ E(G 2 ) ∪ {uv : u ∈ V (G 1 ) and v ∈ V (G 2 )}. In [4,8],…”

“…Vijayan and Kumar [14](2012) obtained some properties of total domination polynomials for cycles. Chaluvaraju et al [4](2014) presented some basic properties of total domination polynomials and graph operations of the union and join of graphs. However, the vertex and edge reduction formulas, which are very important tools to investigate the properties of graph polynomials, are still unknown for total domination polynomials.…”

“…Our results indicate that the number of all total dominating sets of even size and that of odd size can differ by at most 1 for the forest. 4. A direct relation between the vertices of degree 2 and the coefficients of total domina-tion polynomials is given in Theorem 8.…”