On the Energy and Laplacian Energy of the Shadow,Thorn, and Middle Graphs of the Lintang Graph

Abstract

A graph with a transparent initial structure may exhibit markedly different spectral behavior after local or incidence-based transformations. This paper investigates the energy and Laplacian energy of the shadow, one-thorn, and middle graphs derived from the lintang graph Ln = K2 ∨Kn∼ = K2,n. The shadow graph is identified as D2(Ln)∼ = K4,2n, so its adjacency and Laplacian spectra are obtained from the standard spectra of complete bipartite graphs. The one-thorn graph T (Ln) is formed by attaching exactly one pendant vertex to each vertex of Ln, and its spectra are determined through quadratic transformations of the adjacency and Laplacian eigenvalues of the parent graph. The middle graph M (Ln) is analyzed using block matrices, the Schur complement, and rank-one determinant identities. The resulting energy formulas include E(D2(Ln)) = 4√2n and E(T (Ln)) = 2√2n + 4 + 2n, while exact expressions are also obtained for E(M (Ln)) and for the Laplacian energies of all three transformed graphs. In particular, LE(T (Ln)) = (4n/n+2) + 2√2(n−1) + √n2 + 4 + √n2 + 4n + 8. Moreover, D2(Ln) is integral precisely when n = 2k^2 for some positive integer k, whereas T (Ln) and M (Ln) are neither integral nor Laplacian integral for any n ≥ 2. These results show that neighborhood duplication, pendant extension, and incidence expansion produce distinct spectral and arithmetic effects, despite originating from the same complete bipartite base graph.