On the distance spectra of m-generation n-prism graph

Abstract

AbstractThe distance matrix of a simple connected graph G is [Formula: see text] where dij is the length of a shortest path between the ith and jth vertices of G. Eigenvalues of D(G) are called the distance eigenvalues of G. The m-generation n-prism graph or (m, n)-prism graph can be defined in an iterative way where [Formula: see text]-prism graph is an n-vertex cycle. In this paper, we first find the number of zero eigenvalues of the distance matrix of a (m, n)-prism graph. Next, we find some quotient matrix that contains m nonzero distance eigenvalues of a (m, n)-prism graph. Our next result gives the rest of the nonzero distance eigenvalues of a (m, n)-prism graph in terms of distance eigenvalues of a cycle. Finally, we find the characteristic polynomial of the distance matrix of a (m, n)-prism graph. Applying this result, we provide the explicit distance eigenvalues of a [Formula: see text]-prism graph.

Keywords

• Distance matrix
• distance eigenvalues
• path
• cycle
• prism graph
• 05C50