On graphs with the maximum number of spanning trees

• Published in: Random structures & algorithms Vol. 9; no. 1-2; pp. 177 - 192
• Main Author: Kelmans, Alexander K.
• Format: Journal Article
• Language: English
• Published: New York Wiley Subscription Services, Inc., A Wiley Company 01.08.1996
• Subjects: characteristic polynomial; eigenvalues of a matrix; graph; Laplacian polynomial; maximization problem; number of spanning trees; partial order; random graph
• ISSN: 1042-9832; 1098-2418
• DOI: 10.1002/(SICI)1098-2418(199608/09)9:1/2<177::AID-RSA11>3.0.CO;2-L

Let Gmn denote the set of simple graphs with n vertices and m edges, t(G) the number of spanning trees of a graph G, and F ≥ H if t(Ks\E(F)) ≥ t(Ks\E(H)) for every s ≥ max{v(F), v(H)}. We give a complete characterization of ≥‐maximal (maximum) graphs in Gmn subject to m ≤ n. This result contains, in particular, a complete characterization of those graphs in Gkn that have the maximum number of spanning trees among all graphs in Gkn subject to n(n − 1)/2 − n ≤ k ≤ n(n − 1)/2. We discuss and use properties of the Laplacian polynomial L(λ, G) of a graph G [8]. Because of relation t(Ks\E(Gn)) = ss‐n‐2 L(s, Gn) [8], the main result of the paper can also be interpreted as a characterisation of those graphs in Gmn, m ≤ n, having the maximum. Laplacian polynomial for every integer λ ≥ n. We also give an overview of some known approaches and results on this problem. © 1996 John Wiley & Sons, Inc.