Augmented canonical forms and factorization of graphs

The main aim of this paper is to generalize the previously developed canonical forms by different types of alterations and/or augmentations. These forms provide suitable tools for the decomposition of the matrices of special patterns and lead to efficient aproaches for calculating their eigenvalues. Methods were developed for factorization of symmetric graphs into smaller subgraphs using decompostion and healings (Commun. Numer. Meth. Engng 2003; 19:125–136, 2004; 20:133–146, 2004; 20:889–910; Comput. Struct. 2004; 18:2229–2240). Further factorization was possible only when further symmetries were available. The main drawback was associated with non-decomposiblity of some of the factors in early stages of decomposition, and in particular healing which often destroyed the symmetry. In this paper, the mathematical concepts related to certain symmetric and unsymmetric graphs are developed, making refined factorization of the graphs feasible. Examples are included to show the simplicity of the operations being involved.