Abstract
The connectivity index was introduced by Randić (J. Am. Chem. Soc. 97(23):6609–6615, 1975) and was generalized by Bollobás and Erdös (Ars Comb. 50:225–233, 1998). It studies the branching property of graphs, and has been applied to studying network structures. In this paper we focus on the general sum-connectivity index which is a variant of the connectivity index. We characterize the tight upper and lower bounds of the largest eigenvalue of the general sum-connectivity matrix, as well as its spectral diameter. We show the corresponding extremal graphs. In addition, we show that the general sum-connectivity index is determined by the eigenvalues of the general sum-connectivity Laplacian matrix.
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