Asian Research Journal of Mathematics

This study examines the spectral properties of graphs formed by identifying two distinct vertices of a path, with particular emphasis on the singularity and nullity of the adjacency matrix. The analysis considers paths of even and odd order and distinguishes between pendant and non-pendant vertices, as well as core and noncore vertices where these classifications apply. Established results on the adjacency spectra of cycles and paths, vertex null spread, and the characteristic polynomial of graph coalescence are used to determine the effect of vertex identification. For even-order paths, the results classify the fused graphs according to whether the resulting cycle has even or odd length and, in relevant cases, according to the distance between the fused vertex and its nearest pendant vertex. For odd-order paths, the classification depends on whether the identified vertices are both core vertices, both noncore vertices, or one of each. The stated theorems distinguish nonsingular fused graphs from those with nullity one or two under these configurations. Illustrative constructions involving paths of orders eight and nine accompany the theoretical arguments and show how the conditions apply to specified vertex pairs. Overall, the analysis provides a case-based description of how vertex selection and cycle length govern the occurrence and multiplicity of the zero-adjacency eigenvalue in fused graphs of paths.