Abstract
Let Q(G) be the graph derived from G by inserting a new vertex into every edge of G and by connecting edges of these new vertices that are on adjacent edges of G, and Q-double join of G, G 1 and G 2 denoted by G Q v{G 1 , G 2 }, is the graph derived from Q(G), G 1 and G 2 , by connecting every old-vertex V (G) of Q(G) with every vertex of G 1 and every new-vertex I(G) of Q(G) with every vertex of G 2 . In this paper, we mainly considered the resistance distance and Kirchhoff index of Q-double join graphs G Q v {G 1 , G 2 } of an r-regular graph G and any two graphs of G 1 and G 2 .
Highlights
Motivated by all of the above works, we discuss the resistance distance and Kirchhoff index of Q-double join graphs
Let G = (V, E) be a simple graph whose vertex set V (G) = {1, 2, · · ·, n} and edge set E(G) = {e1, e2, · · ·, em}
Proof: With an advisable labeling of the vertices of the graph GQ ∨ {G1, G2}, we can obtain the Laplacian matrix of GQ ∨ {G1, G2} as
Summary
Motivated by all of the above works, we discuss the resistance distance and Kirchhoff index of Q-double join graphs. W. Wang et al.: Resistance Distance and Kirchhoff Index of Q-Double Join Graphs Definition 2 [19]: Let G be a connected graph with n vertices and m edges. Let G1 and G2 be two graphs with n1 and n2 vertices, respectively.
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