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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 Rodr\'iguez [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 Rodr\'iguez. Then, we generalize the previous construction using the key concept of trace-minimal graph introduced by \'Abrego et al. [Linear Algebra Appl. 412 (2006) 161--221]. Finally, as a consequence, we construct infinitely many new t-optimal regular graphs.