Resistance distance in wheels and fans R. B. Bapat ∗ Somit Gupta † February 24, 2009 Abstract: The wheel graph is the join of a single vertex and a cycle, while the fan graph is the join of a single vertex and a path. The resistance distance between any two vertices of a wheel and a fan are obtained. The resistances are related to Fibonacci numbers and generalized Fibonacci numbers. The deriva- tion is based on evaluating determinants of submatrices of the Laplacian matrix. A combinatorial argument is also illustrated. A connectionwith the problem of squaring a rectangle is described. 1 Introduction We consider graphs which have no loops or parallel edges. Thus a graph G = (V (G),E(G)) consists of a finite set of vertices, V (G), and a set of edges, E(G), each of whose elements is a pair of distinct vertices. A weighted graph is a graph in which each edge is assigned a positive number, called its weight. We will assume familiarity with basic graph-theoretic notions, see, for example, Bondy and Murty [4]). Given a graph, one associates a variety of matrices with the graph. Some of the important ones will be defined now. Let G be a graph with V (G)= {1,...,n},E(G)= {e 1 ,...,e m }. The adjacency matrix A(G) of G is an n × n * Indian Statistical Institute, New Delhi, 110016, rbb@isid.ac.in; The author gratefully acknowledges the support of the JC Bose Fellowship, Department of Science ad Technology, Government of India. † National Institute of Technology Karnataka, Surathkal, Karnataka India, somit.gupta@gmail.com 1