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

Abstract

Let ${\mu }_1,{\mu }_2,\dots ,{\mu }_n$ 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 $\mathrm{S}\mathrm{E}\mathrm{E}(G)=\sum _{i=1}^n{e}^{{\mu }_i}$. 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 $\mathrm{S}\mathrm{E}\mathrm{E}$ and sum-connectivity energy. Moreover, we characterize that the star has maximum sum-connectivity Estrada index among trees on $n$ vertices.