Distance spectral spread of a graph

  • Authors:

  • Guanglong Yu;Hailiang Zhang;Huiqiu Lin;Yarong Wu;Jinlong Shu

  • Affiliations:

  • Department of Mathematics, Yancheng Teachers University, Yancheng, 224002, Jiangsu, China and Department of Mathematics, East China Normal University, Shanghai, 200241, China;Department of Mathematics, East China Normal University, Shanghai, 200241, China and Department of Mathematics, Taizhou University, Taizhou, 317000, Zhejiang, China;Department of Mathematics, East China Normal University, Shanghai, 200241, China;Department of Mathematics, East China Normal University, Shanghai, 200241, China and SMU College of Art and Science, Shanghai Maritime University, Shanghai, 200135, China;Department of Mathematics, East China Normal University, Shanghai, 200241, China

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2012

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Abstract

Let D(G)=(d"i","j)"n"x"n denote the distance matrix of a connected graph G with order n, where d"i"j is equal to the distance between vertices v"i and v"j in G. The value D"i=@?"j"="1^nd"i"j (i=1,2,...,n) is called the distance degree of vertex v"i. Denote by @r(G),@r"n(G) the largest eigenvalue and the smallest eigenvalue of D(G) respectively. The distance spectral spread of a graph G is defined to be S"D(G)=@r(G)-@r"n(G). In this paper, some lower bounds on S"D(G) are given in terms of distance degrees, the largest vertex degree and clique number; the spreads of K"n, K"@?"n"2"@?","@?"n"2"@?, K"n"-"@a@?@aK"1, K"1","n"-"1 are proved to be the least among all connected graphs with n vertices, all bipartite graphs with n vertices, all the graphs with both n vertices and independent number @a, all trees with n vertices respectively.