A formula for the number of spanning trees of a multi-star related graph

Chebyshev polynomials and spanning tree formulas for circulant and related graphs

“…Weinberg's results have also been generalized in [22]. Closed formulae also exist for the cases where S is a star [20], a complete k-partite graph [21], a multi-star [19,25], and so on. The number of spanning trees in the complement graph is investigated in [13,16] when the graph with maximal number of spanning trees is studied.…”

The Number of Spanning Trees in Kn-Complements of Quasi-Threshold Graphs

“…Thus, for both theoretical and practical purposes, we are interested in deriving formulas for the number of spanning trees of classes of graphs of the form K n − H. Many cases have already been examined. For example there exist formulas for the cases when H is a pairwise disjoint set of edges [20], when it is a star [17], when it is a complete graph [1], when it is a path [5], when it is a cycle [5], when it is a multi-star [3,16,22], and so on (see Berge [1] for an exposition of the main results).…”

“…Our proofs are based on a classic result known as the complement spanning-tree matrix theorem [19], which expresses the number of spanning trees of a graph G as a function of the determinant of a matrix that can be easily constructed from the adjacency relation (adjacency matrix, adjacency lists, etc.) of the graph G. Calculating the determinant of the complement spanning-tree matrix seems to be a promising approach for computing the number of spanning trees of families of graphs of the form K n − H, where H posses an inherent symmetry (see [1,3,5,16,22,23]). In our cases, since neither trees nor quasi-threshold graphs possess any symmetry, we focus on their structural and algorithmic properties.…”

Number of spanning trees of some families of graphs generated by a triangle

“…Thus for both theoretical and practical purpose, we interested to deriving formulas for the number of spanning trees of a graph based on its time complexity in order to calculate the formula. Many cases have been examined depending on the choice of G. It has been studied when G is labelled molecular graph [1], when G is a circulant graph [2], when G is a complete multipartite graph [3], when G is a cubic cycle and quadruple cycle graph [4], when G is a threshold graph [5] and so on. A spanning tree of G is a minimal connected subgraph of G that has the same vertex set as G. The number of spanning trees in G, also called, the complexity of the graph, denoted by τ ( G ).…”