Resistance Distance in Tensor and Strong Product of Path or Cycle Graphs Based on the Generalized Inverse Approach

Graph product plays a key role in many applications of graph theory because many large graphs can be constructed from small graphs by using graph products. Here, we discuss two of the most frequent graph-theoretical products. Let and be two graphs. The Cartesian product of any two graphs and is a graph whose vertex set is and if either and or and . The tensor product of and is a graph whose vertex set is and if and . The strong product of any two graphs and is a graph whose vertex set is defined by and edge set is defined by . The resistance distance among two vertices and of a graph is determined as the effective resistance among the two vertices when a unit resistor replaces each edge of . Let and denote a path and a cycle of order , respectively. In this paper, the generalized inverse of Laplacian matrix for the graphs and was procured, based on which the resistance distances of any two vertices in and can be acquired. Also, we give some examples as applications, which elucidated the effectiveness of the suggested method.