From Wheel to Jahangir graphs: spectra and energy under structured edge deletion

Abstract

The energy of a graph, introduced by Gutman, is defined as the sum of the absolute values of the eigenvalues of its adjacency matrix. In this work we study how the energy of the Wheel graph $W_n$ changes when a specific set of spokes is removed. The resulting graphs are precisely the Jahangir graphs $J_{m,n}$, where m divides n, and which may also be viewed as the Cycle graph $C_n$ together with an additional vertex adjacent to certain vertices of $C_n$ in a cyclic pattern. We determine the spectra of $J_{2,n}$, $J_{4,n}$, and $J_{6,n}$ by exploiting a factorization of their characteristic polynomials arising from equitable partitions. This yields explicit closed formulas for their energies. Moreover, under certain conditions, we establish the following monotonicity chain: \[\mathcal{E}(C_n) \;\leq\; \mathcal{E}(J_{6,n}) \;\leq\; \mathcal{E}(J_{4,n}) \;\leq\; \mathcal{E}(J_{2,n}) \;\leq\; \mathcal{E}(W_n).\]

arXiv comment16 pages, 1 figure