Spectra of R-Vertex Join and R-Edge Join of Two Graphs

Abstract

The R-graph R(G) of a graph G is the graph obtained from G by intro- ducing a new vertex ue for each e ∈ E(G) and making ue adjacent to both the end vertices of e. In this paper, we determine the adjacency, Lapla- cian and signless Laplacian spectra of R-vertex join and R-edge join of a connected regular graph with an arbitrary regular graph in terms of their eigenvalues. Moreover, applying these results we construct some non-regular A-cospectral, L-cospectral and Q-cospectral graphs, and find the number of spanning trees.

Keywords

• spectrum
• cospectral graphs
• r-vertex join
• r-edge join
• 05c50