Construction of infinitely many trace-minimal graphs with maximum number of spanning trees
Pablo Romero, Louis Petingi
TL;DR
The paper addresses the problem of maximizing the spanning-tree count $t(G)$ for given $(n,m)$ by extending the Petingi–Rodríguez approach with Ábrego et al.'s trace-minimal framework. It develops sharper bounds for $t(ar{G})$ that incorporate Laplacian-gap sequences $g_k(G)$ and introduces a hierarchical graph class $ cal{S}_{n,m}^{(k+1)}$, enabling a generalized construction of $t$-optimal graphs. A duality between $ cal{L}$-trace-minimal graphs and trace-minimal graphs is established, showing that, under suitable conditions, the complement of a trace-minimal base yields infinitely many $t$-optimal regular graphs. The results provide a systematic pathway to generate new regular $t$-optimal graphs, via $G_0(p,q)$ and related expansions, with concrete infinite families such as the $H_n$ graphs for large $n$ and explicit corollaries drawn from trace-minimal bases.
Abstract
A longstanding problem in spectral graph theory asks for graphs with maximum number of spanning trees among all connected simple graphs with a prescribed number of vertices and edges. Such graphs are called t-optimal graphs. Petingi and Rodríguez [Discrete Math. 244 (2002), 351--373] achieved in finding infinitely many t-optimal graphs. Basically, they reduced the problem of finding t-optimal graphs to the determination of almost-regular graphs with minimum number of induced 3-paths. In this work we revisit the construction of t-optimal graphs given by Petingi and Rodríguez. Then, we generalize the previous construction using the key concept of trace-minimal graph introduced by Ábrego et al. [Linear Algebra Appl. 412 (2006) 161--221]. Finally, as a consequence, we construct infinitely many new t-optimal regular graphs.