On graphs with the maximum number of spanning trees
Proceedings of the seventh international conference on Random structures and algorithms
On the characterization of graphs with maximum number of spanning trees
Discrete Mathematics
Graph Theory With Applications
Graph Theory With Applications
Counting spanning trees in graphs using modular decomposition
WALCOM'11 Proceedings of the 5th international conference on WALCOM: algorithms and computation
| Hi-index | 0.00 |
Let Γ(n,e) denote the class of all simple graphs on n nodes and e edges. The number of spanning trees of a graph G is denoted by t(G). A graph G0 ∈ Γ(n,e) is said to be t-optimal if t(G0) ≥ t(G) for all G ∈ Γ(n,e). The problem of characterizing t-optimal graphs for arbitrary n and e is still open, although characterizations of t-optimal graphs for specific pairs (n,e) are known. We introduce a new technique for the characterization of t-optimal graphs, based on an upper bound for the number of spanning trees of a graph G in terms of the degree sequence and the number of induced paths of length two of the complement of G. The technique yields the following new results: (1) Complete, almost-regular multipartite graphs are t-optimal. (2) A complete characterization of t-optimal graphs in Γ(n,e) for n(n - 1)/2 - 3n/2 ≤ e ≤ n(n - 1)/2- n is obtained for n ≥ n0, where n0 can be explicitly determined.