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On the resistance diameter of the Cartesian and lexicographic product of paths

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Abstract

Let G be a graph with vertex set V(G). For \(u,v\in V(G)\), we write \(v\sim u\) if vertices v and u are adjacent. The Cartesian product of G and H, denoted by \(G\Box H\), is the graph with vertex set \(V(G)\times V(H)\), where \((x,u)\sim (y,v)\) if and only if \(x=y\) and \(u\sim v\) in H, or \(x\sim y\) in G and \(u=v\). The lexicographic product of G and H, denoted by G[H], is the graph with vertex set \(V(G)\times V(H)\), where \((x,u)\sim (y,v)\) if and only if \(x\sim y\) in G, or \(x=y\) and \(u\sim v\) in H. The resistance diameter of graph G refers to the maximum resistance distance among all pairs of vertices in G. Let \(P_n\) be the path of n vertices. In this paper, the resistance diameters of \(P_n\Box P_m\) and \(P_n[P_m]\) are studied. Meanwhile, the maximal resistance distance, which is among some pairs of vertices in the lexicographic product of connected graph and orderable graph, is given.

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Acknowledgements

The authors would like to express their sincere gratitude to the anonymous referees for valuable suggestions, which led to great deal of improvement of the original manuscript. Hongbo Hua was supported by the National Natural Science Foundation of China under Grant No. 11971011 and Qing Lan Project of the Jiangsu Province of China. Xiang-Feng Pan was supported by University Natural Science Research Project of Anhui Province under Grant No. KJ2020A0001.

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Authors and Affiliations

  1. School of Mathematical Sciences, Anhui University, Hefei, 230601, Anhui, People’s Republic of China

    Yun-Xiang Li, Si-Ao Xu & Xiang-Feng Pan

  2. Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huai’an, 223003, Jiangsu, People’s Republic of China

    Hongbo Hua

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  1. Yun-Xiang Li
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Correspondence to Xiang-Feng Pan.

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Li, YX., Xu, SA., Hua, H. et al. On the resistance diameter of the Cartesian and lexicographic product of paths. J. Appl. Math. Comput. 68, 1743–1755 (2022). https://doi.org/10.1007/s12190-021-01587-w

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  • DOI: https://doi.org/10.1007/s12190-021-01587-w

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