On the edge cover polynomial of a graph

  • Authors:

  • Saieed Akbari;Mohammad Reza Oboudi

  • Affiliations:

  • Department of Mathematical Sciences, Sharif University of Technology, P.O. Box 11155-9415, Tehran, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19 ...;Department of Mathematics, University of Isfahan, Isfahan, 81746-73441, Iran and School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

  • Venue:
  • European Journal of Combinatorics
  • Year:
  • 2013

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Abstract

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. In this paper we introduce a new graph polynomial. The edge cover polynomial of G is the polynomial E(G,x)=@?"i"="1^me(G,i)x^i, where e(G,i) is the number of edge coverings of G of size i. Let G and H be two graphs of order n such that @d(G)=n2, where @d(G) is the minimum degree of G. If E(G,x)=E(H,x), then we show that the degree sequence of G and H are the same. We determine all graphs G for which E(G,x)=E(P"n,x), where P"n is the path of order n. We show that if @d(G)=3, then E(G,x) has at least one non-real root. Finally, we characterize all graphs whose edge cover polynomials have exactly one or two distinct roots and moreover we prove that these roots are contained in the set {-3,-2,-1,0}.