Extremal Values of Dharwad Index

초록

The Dharwad Index is a graph-theoretic measure defined as the sum of the square roots of the cubic sums of the degrees of all adjacent vertex pairs in a graph. In this paper, we systematically investigate the Dharwad index for trees, determining its extremal values, including the maximum, minimum, second maximum, and second minimum, and identifying the corresponding extremal tree structures. This investigation establishes that the path and star graphs are the unique minimizer and maximizer for trees, and identifies the specific tree and unicyclic structures that achieve the second extremal values and various bounds. In addition, several sharp bounds for the Dharwad index of general graphs are derived, along with a characterization of the extremal graphs attaining these bounds. The study is further extended to unicyclic graphs, where the extremal values of the Dharwad index are analyzed in detail. Overall, this work demonstrates the effectiveness of the Dharwad index in capturing essential structural properties across diverse classes of graphs.

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