University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 17 1 2026 03 01 On the Energy and Nullity of Non-Uniform Path and Cycle Semigraphs 53 75 115419 10.22052/ijmc.2025.257303.2041 EN Serin Elezabeth Joy Department of Mathematics‎, ‎Mar Athanasius College of Engineering (Autonomous)‎, ‎Kothamangalam‎, Ernakulam‎, ‎686666‎, ‎Kerala‎, ‎India Rajesh K‎. ‎ Thumbakara Department of Mathematics‎, ‎Mar Athanasius College (Autonomous)‎, ‎Kothamangalam‎, ‎Ernakulam‎, ‎686666‎, ‎Kerala‎, ‎India Journal Article 2025 07 27 ‎Graph energy‎, ‎originating in H\"uckel molecular orbital theory‎, ‎remains central to mathematical chemistry‎. ‎Motivated by heterogeneous linear and cyclic molecular structures‎, ‎we study non-uniform path and cycle semigraphs‎, ‎where original edges are subdivided by $n_i \ge 1$ middle vertices‎. ‎We show the adjacency matrix decomposes into a symmetric tridiagonal core‎, ‎whose spectrum comprises all non-zero eigenvalues‎, ‎plus zero rows from middle vertices‎. ‎For paths‎, ‎a continuant recurrence for the characteristic polynomial and parity arguments yield spectral symmetry and precise nullity conditions‎. ‎For cycles‎, ‎a wraparound determinant formula characterizes when the spectrum is symmetric about zero and provides exact criteria for the presence and multiplicity of specific zero eigenvalues‎. Consequently, the energy of each semigraph equals the energy of its core matrix‎, ‎yielding clean expressions for energy and nullity from the $\{n_i\}$ parameters‎. Uniform cases arise as immediate corollaries and are consistent with spectral invariants in chemically inspired models. https://ijmc.kashanu.ac.ir/article_115419_a3ed3d892b4e24a2ba550a765a08cae5.pdf