Theoretical and Computational Methods to Resistance Distances in Novel Graphs Operations

Motivated by the recent research on the computation of resistance distance, this paper aims to compute resistance distance in two classes of graphs, which are generated by three graphs. In fact, they are
$G_{1}(\vee _{H})G_{2}$
and
$G_{0}^{S}\lhd (G_{1}^{V}\cup G_{2}^{E})$
. In this paper, we first give the
$\{1\}$
-inverses of the Laplacian matrix of
$G_{1}(\vee _{H})G_{2}$
and
$G_{0}^{S}\lhd (G_{1}^{V}\cup G_{2}^{E})$
by calculation. Then connected with the relationship between resistance distance and the
$\{1\}$
-inverses of the Laplacian matrices, we would obtain resistance distance in
$G_{1}(\vee _{H})G_{2}$
and
$G_{0}^{S}\lhd (G_{1}^{V}\cup G_{2}^{E})$
. In addition, we finally list two examples to illustrate the efficiency of our proposed method.