Reciprocal complementary distance equienergetic graphs. (English) Zbl 1352.05118
Summary: The reciprocal complementary distance (RCD) matrix of a graph \(G\) is defined as RCD(G)=\([rc_{ij}]\), in which \(rc_{ij}=\frac{1}{1+D-d_{ij}}\) if \(i\neq j\) and \(rc_{ij}=0\) if \(i=j\), where \(D\) is the diameter of \(G\) and \(d_{ij}\) is the distance between the \(i\)th and \(j\)th vertex of \(G\). The RCD-energy [RCDE(\(G\))] of \(G\) is defined as the sum of the absolute values of the eigenvalues of RCD-matrix of \(G\). Two graphs \(G_1\) and \(G_2\) are said to be RCD-equienergetic if RCDE(\(G_1\))=RCDE(\(G_2\)). In this paper, we obtain the RCD-eigenvalues and RCD-energy of the join of certain regular graphs and thus construct the non-RCD-cospectral, RCD-equienergetic graphs on \(n\) vertices, for all \(n\geq 9\).
Keywords:
reciprocal complementary distance eigenvalues; reciprocal complementary distance energy; join of graphsReferences:
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