Dense Graph Homomorphisms of Co spectral Rigid Circuit Graphs are Cycle Decompositions

Abstract

Understanding the structure of graphs is essential for progress in various areas of graph
theory and its applications. This paper focuses on a specific structural property across multiple
classes of graphs. Graphs containing a clique of size r(i.e., Kr-cospectralchordal dense graphs) are
particularly significant in external graph theory. While many results have been established
regarding dense triangle-free graphs, less is known about dense Kr-cospectralchordal dense graphs
for r  4. The conditions required to decompose the entire graph of odd order into cycles of a fixed
even length and the complete graph of even order minus 1-factor into cycles of a fixed odd length
are also demonstrated to be sufficient. This paper highlights key findings related to cores,
homomorphisms and cycle decomposition.