Chordal 2‐Connected Graphs and Spanning Trees

Abstract

Abstract:We present a transformation on a chordal 2-connected simple graph that decreases the number of spanning trees. Based on this transformation, we show that for positive integers n, m with n(n − 1)/2 ≥ m ≥ 2n − 3, the threshold graph Q n,m having n vertices and m edges that consists of an (n − k )-clique and k − 1 vertices of degree 2 is the only graph with the fewest spanning trees among all 2-connected chordal graphs on n vertices and m edges.

On q-connected chordal graphs with minimum number of spanning trees

“…For 2-connected graphs, such a series of shift transformations that produces a 2-connected threshold graph does not always exists [2]. For chordal q-connected graph G, however, we show in the next section that such a series of shift transformations does exist, and it transforms G to q-connected chordal graph H.…”

“…Subsequently, we showed in [3] that specific subset of graphs minimizes the number of spanning trees and represents all such graphs. Finally, in [2] it was shown that there is a unique threshold G ∈ S(2, n, m) such that t(G) < t(H) for any H that satisfies H ∈ S(2, n, m) and H ≇ G. In this paper, we extend this latest result to G ∈ S(q, n, m), where n−k 2 ≥ q ≥ 2.…”

“…The following three theorems for 2-connected chordal graphs pertaining to shif t1(G, v, w) and shif t2(G, v, w) were proved in [2].…”

“…Theorem 6 [2]. For any chordal 2-connected G that is not a threshold graph, there is a series of shift transformations, consisting of shift1 and shift2, that produces 2-connected threshold graph H, with the same numbers of vertices and edges, such that t(H) < t(G).…”