Use of symmetry plane and subsequent subtraction for obtaining eigenspectra of some complicated graphs in analytical forms

Combining symmetry plane factorization and subsequent subtraction a graph theoretical procedure has been presented to obtain analytical expressions of eigenspectra for the graphs of linear chains which have one or two terminal edge(s) of weight 2k and all other edge(s) of equal weight k. This procedure is illustrated with both the linear chains of same vertex-weight (h) and alternant vertex-weight (h1, h2). These expressions of eigenspectra have been used to calculate eigenspectra of linear polyacenes or methylene substituted linear polyacenes with two or four edges of weight 2. The method of subtraction have also been directly exploited to obtain the expressions of eigenspectra of the said linear polyacenes graphs, stack graphs with terminal edges of weight 2 in one side or both the sides and reciprocal graphs with one or two terminal base-edges of weight 2. Some subspectral and or isospectral relations among the graphs have also been followed from the respective expressions of eigenspectra. Edge weighted graphs are generally factored subgraphs. Sometimes these graphs can be viewed as molecules or radicals where the vertices corresponding to the weighted edge or edges are occupied by the heteroatoms, i.e. other than carbon atoms.