In view of the wide application of resistance distance, the computation of resistance distance in various graphs becomes one of the main topics. In this paper, we aim to compute resistance distance in H-join of graphs G(1), G(2), ... , G(k). Recall that H is an arbitrary graph with V(H) =3D {1, 2, ... , k}, and G(1), G(2), ... , G(k) are disjoint graphs. Then, the H-join of graphs G(1), G(2), ... , G(k), denoted by V-H{G(1), G(2), ... , G(k)}, is a graph formed by taking G(1), G(2), ... , G(k) and joining every vertex of G(i) to every vertex of G(j) whenever i is adjacent to j in H. Here, we first give the Laplacian matrix of V-H {G(1), G(2), ... , G(k)}, and then give a {1}-inverse L(V-H {G(1), G(2), ... , G(k))({)(1)(}) or group inverse L(V-H{G(1), G(2), ... , G(k)})(# )of L(V-H{G(1), G(2), ... , G(k)). It is well know that, there exists a relationship between resistance distance and entries of {1}-inverse or group inverse. Therefore, we can easily obtain resistance distance in V-H{G(1), G(2) , ... ,G(k)}. In addition, some applications are presented in this paper.

In this note, we describe a seemingly new approach to the complex representation theory of the wreath product \(G\wr S_d\), where G is a finite abelian group. The approach is motivated by an appropriate version of Schur–Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of \(G\wr S_d\). This directly implies a classification of simple modules. As an application, we get a Gelfand model for \(G\wr S_d\) from the classical involutive Gelfand model for the symmetric group. We describe the Schur–Weyl duality which motivates our approach and relate it to various Schur–Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type \(G(\ell ,k,d)\).

In this note, we describe a seemingly new approach to the complex representation theory of the wreath product , where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of . This directly implies a classification of simple modules. As an application, we get a Gelfand model for from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type G(l, k, d).