Bounds and extremal graphs for Harary energy. (English) Zbl 1493.05183
Summary: Let \(G\) be a connected graph of order \(n\) and let \(\mathrm{RD}(G)\) be the reciprocal distance matrix (also called Harary matrix) of the graph \(G\). Let \(\rho_1\geq \rho_2\geq\cdots\geq \rho_n\) be the eigenvalues of the reciprocal distance matrix \(\mathrm{RD}(G)\) of the connected graph \(G\) called the reciprocal distance eigenvalues of \(G\). The Harary energy \(\mathrm{HE}(G)\) of a connected graph \(G\) is defined as sum of the absolute values of the reciprocal distance eigenvalues of \(G\), that is, \( \mathrm{HE}(G)= \sum_{i = 1}^n| \rho_i|\). In this paper, we establish some new lower and upper bounds for \(\mathrm{HE}(G)\), in terms of different graph parameters associated with the structure of the graph \(G\). We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of \(k\) largest adjacency eigenvalues of a connected graph.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C12 | Distance in graphs |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
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