The {1}-inverse of the Laplacian of subdivision-vertex and subdivision-edge coronae with applications

The subdivision graph S(G) of a graph G is the graph obtained by inserting a new vertex into every edge of G. Let and be two vertex disjoint graphs. The subdivision-vertex corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of . The subdivision-edge corona of and , denoted by , is the graph obtained from and copies of , all vertex-disjoint, by joining the ith vertex of to every vertex in the ith copy of , where is the set of inserted vertices of . In this paper, -inverse for the Laplacian matrices of graphs and are proposed, based on which the explicit resistance distance can be obtained for the two-vertex resistance between arbitrary vertices in the graphs.

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