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On the edge cover polynomial of a graph

Akbari, S ; Sharif University of Technology | 2013

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  1. Type of Document: Article
  2. DOI: 10.1016/j.ejc.2012.05.005
  3. Publisher: 2013
  4. Abstract:
  5. 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=1me(G,i)xi, 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 δ(G)≥n2, where δ (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 δ (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 }
  6. Keywords:
  8. Source: European Journal of Combinatorics ; Volume 34, Issue 2 , 2013 , Pages 297-321 ; 01956698 (ISSN)
  9. URL: http://www.sciencedirect.com/science/article/pii/S0195669812000996