The connectivity index was introduced by Randi´c (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.