Characteristic Polynomial Analysis on Matrix Representations of Graphs

Abstract

Matrix representations for graphs play an important role in combinatorics. In this paper, we investigate four matrix representations for graphs and carry out an characteristic polynomial analysis upon them. The first two graph matrices are the adjacency matrix and Laplacian matrix. These two matrices can be obtained straightforwardly from graphs. The second two matrix representations, which are newly introduced [9][3], are closely related with the Ihara zeta function and the discrete time quantum walk. They have a similar form and are established from a transformed graph, i.e. the oriented line graph of the original graph. We make use of the characteristic polynomial coefficients of the four matrices to characterize graphs and construct pattern spaces with a fixed dimensionality. Experimental results indicate that the two matrices in the transformed domain perform better than the two matrices in the original graph domain whereas the matrix associated with the Ihara zeta function is more efficient and effective than the matrix associated with the discrete time quantum walk and the remaining methods.