On the Sum-Connectivity Spectral Radius and Sum-Connectivity Estrada Index of Graphs

Abstract

Let \(\mu_1, \mu_2, \ldots, \mu_n\) be the eigenvalues of the sum-connectivity matrix of a graph \(G\). The sum-connectivity spectral radius of \(G\) is the largest eigenvalue of its sum-connectivity matrix, and the sum-connectivity Estrada index of \(G\) is defined as \(\mathrm{SEE}(G) = \sum_{i=1}^{n} e^{\mu_i}\). In this paper, we obtain some results about the sum-connectivity spectral radius of graphs. In addition, we give some upper and lower bounds on the sum-connectivity Estrada index of graph \(G\), as well as some relations between \(\mathrm{SEE}\) and sum-connectivity energy. Moreover, we characterize that the star has maximum sum-connectivity Estrada index among trees on \(n\) vertices.