On the Number of Spanning Trees of Multi-Star Related Graphs

Abstract In this paper we compute the number of spanning trees of a specific family of graphs using techniques from linear algebra and matrix theory. More specifically, we consider the graphs that result from a complete graph K n after removing a set of edges that spans a multi-star graph K m ( a 1 , a 2 ,…, a m ). We derive closed formulas for the number of spanning trees in the cases of double-star ( m = 2), triple-star ( m = 3), and quadruple-star ( m = 4). Moreover for each case we prove that the graphs with the maximum number of spanning trees are exactly those that result when all the a i s are equal.