On the Number of Edges for Chessboard Graphs in Higher Dimensions

Abstract

In this paper, we consider chessboard graphs in higher dimensions and the number of edges of their corresponding graphs. First, we solve for the number of edges for some of the chessboard graphs of 3 dimensions. Then, we obtain results for the number of edges for higher dimensional chessboard graphs with all dimensions sizes being the same. For the higher dimensional queen’s graph, a double series solution for the number of edges was found - given both the dimension size and the number of dimensions. This solution didn’t reduce readily to a closed form solution. For the higher dimensional bishop’s graph, a series solution was found that involves the generalized Harmonic number. This solution series also didn’t readily reduce to a closed form solution. Results for the number of edges for two types of higher dimensional knights’ graphs were also found. Finally, a series solution for the number of edges for the higher dimensional king’s graph was found as well. To reduce this series solution to a closed form solution, the Wolfram Alpha series calculator was utilized.