Abstract
We define a correlated random walk (CRW) induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph $G$, and present a formula for the characteristic polynomial of the transition probability matrix of this CRW by using a determinant expression for the generalized weighted zeta function of $G$. As an application, we give the spectrum of the transition probability matrices for the CRWs induced from the Grover matrices of regular graphs and semiregular bipartite graphs. Furthermore, we consider another type of the CRW on a graph.
Highlights
Zeta functions of graphs started from the Ihara zeta functions of regular graphs by Ihara [7]
Bass [1] presented another determinant expression for the Ihara zeta function of an irregular graph by using its adjacency matrix
Konno and Sato [11] obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a graph
Summary
Zeta functions of graphs started from the Ihara zeta functions of regular graphs by Ihara [7]. Hashimoto [5] generalized Ihara’s result on the Ihara zeta function of a regular graph to an irregular graph, and showed that its reciprocal is again a polynomial by a determinant containing the edge matrix. Konno and Sato [11] obtained a formula of the characteristic polynomial of the Grover matrix by using the determinant expression for the second weighted zeta function of a graph. We introduce a new correlated random walk induced from the time evolution matrix (the Grover matrix) of the Grover walk on a graph, and present a formula for the characteristic polynomial of its transition probability matrix.
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