Ḣ-Join operation of signed graphs constrained by indexing maps

Department of Mathematics and Applications R. Caccioppoli, University of Naples Federico II, Naples, Italy

^* Corresponding author: callum.huntington@unina.it

Received: 18 March 2025
Accepted: 8 July 2025

Abstract

Let H be a graph of order k and let F = {G₁, G₂, . . . , G_k} be a family of k graphs. Then the H-join of the family F is obtained by replacing each vertex v_i of H with the graph G_i of F and respecting the adjacencies existing in H. To generalise this graph operation we consider a signed variant with the addition of fixing m ∈ N and introducing indexing maps to define the Ḣ_m-join. Once having done so we can determine the characteristic polynomials and spectra of the compound graphs produced as pertaining to many graph matrices, such as the adjacency, Laplacian, universal adjacency, net Laplacian, and A_α. Furthermore we show that the Ḣ_m-join remains stable under switching of Ḣ.