Complementary distance spectra and complementary distance energy of line graphs of regular graphs. (English) Zbl 1463.05352
Summary: The complementary distance (CD) matrix of a graph \(G\) is defined as \(CD(G) = [c_{ij}]\), where \(c_{ij} = 1+D-d_{ij}\) if \(i \neq j\) and \(c_{ij} = 0\), otherwise, where \(D\) is the diameter of \(G\) and \(d_{ij}\) is the distance between the vertices \(v_i\) and \(v_j\) in \(G\). The \(CD\)-energy of \(G\) is defined as the sum of the absolute values of the eigenvalues of \(CD\)-matrix. Two graphs are said to be \(CD\)-equienergetic if they have same \(CD\)-energy. In this paper we show that the complement of the line graph of certain regular graphs has exactly one positive \(CD\)-eigenvalue. Further we obtain the \(CD\)-energy of line graphs of certain regular graphs and thus constructs pairs of \(CD\)-equienergetic graphs of same order and having different \(CD\)-eigenvalues.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C12 | Distance in graphs |
| 05C76 | Graph operations (line graphs, products, etc.) |
Keywords:
complementary distance eigenvalues; adjacency eigenvalues; line graphs; complementary distance energyReferences:
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