On trees with extremal extended spectral radius

Abstract: Let G be a simple connected graph with n vertices, and let d_i be the degree of the vertex v_i in G. The extended adjacency matrix of G is defined so that the ij-entry is 1/2(d_i/d_j+d_j/d_i) if the vertices v_i and v_j are adjacent in G, and 0 otherwise. This matrix was originally introduced for developing novel topological indices used in the QSPR/QSAR studies. In this paper, we consider extremal problems of the largest eigenvalue of the extended adjacency matrix (also known as the extended spectral radius) of trees. We show that among all trees of order n>= 5, the path Pn(resp., the star Sn) uniquely minimizes (resp., maximizes) the extended spectral radius. We also determine the first five trees with the maximal extended spectral radius.