Undirected simple connected graphs with minimum number of spanning trees

Abstract

a b s t r a c tWe show that for positive integers n, m with n(n−1)/2 ≥ m ≥ n−1, the graph L n,m having n vertices and m edges that consists of an (n−k)-clique and k−1 vertices of degree 1 has the fewest spanning trees among all connected graphs on n vertices and m edges. This proves Boesch's conjecture [F.T. Boesch, A. Satyanarayana, C.L. Suffel, Least reliable networks and reliability domination, IEEE Trans.

Chordal 2‐Connected Graphs and Spanning Trees

“…Secondly, based on our restricted shift transformation we prove that there is the unique 2-connected threshold graph that minimizes the number of spanning trees over all chordal 2-connected graphs with the same number of vertices and edges. A comparable result has been obtained in [1] for undirected graphs, where specific graph L n,m generalized Kelmans et al result [4] (Kelmans et al proved that a complete graph with deleted star graph minimizes the number of spanning trees) and has been shown to minimize the number of spanning trees over all connected graphs with n vertices and m edges. However, in that case L n,m was not unique as opposed to Kelmans et al result. In the next section, we present some prior work and results that will be useful in this article.…”

“…In this article, we employ shifting transformation shift(G,v,w) on undirected graphs [1,2,8]. Let G = (V, E ) be an undirected graph and, for a vertex v of G, let N(v) denote the vertices that are neighbors to v. We say that vertex v dominates vertex w if every neighbor of w is also a neighbor of v, so N(w) ⊆ N(v).…”

“…It is known that if shift(G,v,w) = G for all v, w, then G is a threshold graph [1,7]. It is the graphs H = H(n; d 1 , d 2 , .…”

“…It was shown in [1,2,7] that every connected graph G can be transformed into a threshold graph H using a series of shift(G,v,w) transformations. Consequently:…”

Nonexistence of optimal graphs for all terminal reliability

“…Previous focus has been exclusively on least optimal simple graphs. It was conjectured by Boesch et al that a least optimal simple graph has the maximum possible number of 2‐connected components, and Bogdanowicz proved that the simple graph with the least number of spanning trees is , which consists of an ‐clique, joined to k – 1 leaves plus one other vertex of degree , where k is the least integer such that . So if there is a least optimal simple graph on n vertices and m edges, then is a candidate.…”