On Walk Counts and Complexity of General Graphs

Motivation. This report was motivated by a recent work of Gutman, Rucker and Rucker on walks in simple molecular graphs, i.e., graphs without multiple edges and loops. Method. Three methods for counting walks in general graphs, i.e., graphs with multiple bonds and loops, are presented: (i) graphical method based on the Morgan summation procedure, (ii) method based on augmented adjacency matrices of higher orders and (iii) method based on eigenvalues and eigenvectors of augmented adjacency matrices of higher orders. They represent extensions of the methods discussed previously in the literature for simple graphs. Results. The total walk count (twc) was used as a measure for complexity of general graphs. It is shown that twc indices increase with size, branching, cyclicity, the number of loops and multiple bonds, and decrease with symmetry of general graphs. Conclusions. The total walk count appears to be a valuable tool to account for complexity for several types of molecular graphs.

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