The Complexity of Wheel Graphs with Multiple Edges and Vertices

Abstract

In this paper, we focus on calculate the number of spanning trees of the general wheel graphs, which meansthe original wheel graphs adding large amount of vertices and edges. Particularly, we introduce the C-graphand deduce a new equation that computing the spanning trees by removing C-graphs instead of edges.In Addition, we test our results by Kirchhoff’s matrix-tree theorem in some simple cases and provide thetree entropy of the general wheel graphs. Finally, we analyse the relation between the wheel graph anddouble-wheel graphs and propose the idea of calculating the spanning trees of double-wheel graphs.