A proof of unimodality on the numbers of connected spanning subgraphs in an n-vertex graph with at least ⌈(3-22)n2+n-7-2222⌉ edges

  • Authors:

  • Peng Cheng;Shigeru Masuyama

  • Affiliations:

  • Nagoya Gakuin University, Nagoya, 456-8612, Japan;Toyohashi University of Technology, Toyohashi, 441-8580, Japan

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

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Abstract

Consider a connected undirected simple graph G=(V,E) with n vertices and m edges, and let N"i denote the number of connected spanning subgraphs with i(n-1@?i@?m) edges in G. Two well-known open problems are whether N"n"-"1,N"n,...,N"m is unimodal (posed by Welsh (1971) [21]), and whether it is log concave (posed by Mason (1972) [13]). Here, a sequence of real numbers a"0,a"1,...,a"m is said to be unimodal if there is an index i(0@?i@?m) such that a"0@?a"1@?...@?a"i=a"i"+"1=...=a"m, and log concave if a"i^2=a"i"-"1a"i"+"1 for all indices i(0