Abstract: Let G be a graph of order n. The second stage adjacency matrix of G is the symmetric n × n matrix for which the ijth entry is 1 if the vertices vi and vj are of distance two; otherwise 0. The sum of the absolute values of this second stage adjacency matrix is called the second stage energy of G. In this paper we investigate a few properties and determine some upper bounds for the largest eigenvalue.
Abstract: The Detour matrix (DD) of a graph has for its ( i , j)
entry the length of the longest path between vertices i and j. The
DD-eigenvalues of a connected graph G are the eigenvalues for its
detour matrix, and they form the DD-spectrum of G. The DD-energy
EDD of the graph G is the sum of the absolute values of its DDeigenvalues.
Two connected graphs are said to be DD- equienergetic
if they have equal DD-energies. In this paper, the DD- spectra of a
variety of graphs and their DD-energies are calculated.