The spectra of signed graphs obtained by $\dot{H}$-(generalized) join operation

Abstract

For a signed graph $\dot{H}$ with h vertices and a family of h signed graphs $\{\dot{G}_{1}, \dot{G}_{2}, \ldots , \dot{G}_{h}\}$, we consider the $\dot{H}$-join $\bigvee _{\dot{H}}\{\dot{G}_{1}, \dot{G}_{2}, \ldots ,\dot{G}_{h}\}$ and the $\dot{H}$-generalized join $\bigvee _{\dot{H}, {\mathcal {S}}}\{\dot{G}_{1}, \dot{G}_{2}, \ldots ,\dot{G}_{h}\}$ of $\dot{G}_{1}, \dot{G}_{2}, \ldots ,\dot{G}_{h}$. In this paper, we characterize the net Laplacian spectrum, the normalized Laplacian spectrum and the universal adjacency spectrum of every signed graph obtained by the $\dot{H}$-join operation on disjoint signed graphs. Moreover, we determine the N-characteristic polynomial, the net Laplacian characteristic polynomial and the corresponding spectra of every signed graph obtained by the $\dot{H}$-generalized join operation, where $N(\dot{G})=A(\dot{G})-tD(G)$ with $t\in {\mathbb {R}}$.