Bounds on the index of the signless Laplacian of a graph

  • Authors:

  • Carla Silva Oliveira;Leonardo Silva de Lima;Nair Maria Maia de Abreu;Pierre Hansen

  • Affiliations:

  • School of Statistical Sciences, Rio de Janeiro, Brazil;Federal Center of Technological Education, Rio de Janeiro, Brazil;Federal University of Rio de Janeiro, Brazil;GERAD, Canada and HEC Montréal, Canada

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2010

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Abstract

Let G=(V,E) be a simple, undirected graph of order n and size m with vertex set V, edge set E, adjacency matrix A and vertex degrees @D=d"1=d"2=...=d"n=@d. The average degree of the neighbor of vertex v"i is m"i=1d"i@?"j"="1^na"i"jd"j. Let D be the diagonal matrix of degrees of G. Then L(G)=D(G)-A(G) is the Laplacian matrix of G and Q(G)=D(G)+A(G) the signless Laplacian matrix of G. Let @m"1(G) denote the index of L(G) and q"1(G) the index of Q(G). We survey upper bounds on @m"1(G) and q"1(G) given in terms of the d"i and m"i, as well as the numbers of common neighbors of pairs of vertices. It is well known that @m"1(G)@?q"1(G). We show that many but not all upper bounds on @m"1(G) are still valid for q"1(G).