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Sharp bounds on the distance spectral radius and the distance energy of graphs. (English) Zbl 1165.05019

Summary: The \(D\)-eigenvalues \(\{\mu _{1},\mu _{2},\ldots ,\mu _p\}\) of a graph \(G\) are the eigenvalues of its distance matrix \(D\) and form the \(D\)-spectrum of \(G\) denoted by spec\(_D(G)\). The greatest \(D\)-eigenvalue is called the \(D\)-spectral radius of \(G\) denoted by \(\mu _{1}\). The \(D\)-energy \(E_D(G)\) of the graph \(G\) is the sum of the absolute values of its \(D\)-eigenvalues. In this paper we obtain some lower bounds for \(\mu _{1}\) and characterize those graphs for which these bounds are best possible. We also obtain an upperbound for \(E_D(G)\) and determine those maximal \(D\)-energy graphs.

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
05C12 Distance in graphs
05C35 Extremal problems in graph theory
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