Resistance distance in some classes of rooted product graphs obtained by Laplacian generalized inverse method

Abstract

In mathematics, a graph product is a binary operation on a graph. Graph products have been extensively researched and have many important applications in many fields. Here we discuss one graph-theoretical product. Let H be a labeled graph on n vertices and let G be a rooted graph. Denote by H G the graph obtained by identifying the root vertex of the ith copy of G with the ith vertex of H. H G is called by the rooted product of H by G [C. Godsil, B. D. McKay, A new graph product and its spectrum, Bull. Aust. Math. Soc. 18 (1978)]. The resistance distance between two vertices i and j of a graph G is defined as the effective resistance between the two vertices when a unit resistor replaces each edge of G. Let H_k;n, C_m, S_k, P_k and K_u be the Harary, cycle, star, path and complete graphs respectively. In this paper, the symmetric {1}-inverses of Laplacian matrices for graphs (H_k;n ○ C_m), (H_k;n K_u), (C_n ○ S_k) and (C_n ○ P_k) are studied, based on which the resistance distances of any two vertices in these graphs can be obtained. In addition, some examples are provided as applications that illustrate the functionality of the suggested method.

Subject Classification: (2020):

• 05C50
• 05C12
• 15A09

Keywords:

• Laplacian matrix
• Generalized inverse
• Moore-Penrose inverse
• Schur complement
• Resistance distance