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 G1 and G2 be two graphs. The Cartesian product G1□G2 of any two graphs G1 and G2 is a graph whose vertex set is VG1□G2=VG1×VG2 and a1,a2b1,b2∈EG1□G2 if either a1=b1 and a2b2∈EG2 or a1b1∈EG1 and a2=b2. The tensor product G1×G2 of G1 and G2 is a graph whose vertex set is VG1×G2=VG1×VG2 and a1,a2b1,b2∈EG1×G2 if a1b1∈EG1 and a2b2∈EG2. The strong product G1⊠G2 of any two graphs G1 and G2 is a graph whose vertex set is defined by VG1⊠G2=VG1×VG2 and edge set is defined by EG1⊠G2=EG1□G2∪EG1×G2. The resistance distance among two vertices u and v of a graph G is determined as the effective resistance among the two vertices when a unit resistor replaces each edge of G. Let Pn and Cn denote a path and a cycle of order n, respectively. In this paper, the generalized inverse of Laplacian matrix for the graphs Pn1×Cn2 and Pn1⊠Pn2 was procured, based on which the resistance distances of any two vertices in Pn1×Cn2 and Pn1⊠Pn2 can be acquired. Also, we give some examples as applications, which elucidated the effectiveness of the suggested method.