On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices

This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index SEIa and the general zeroth-order Randić index Rα0. The minimum values of SEIa and Rα0 in the class of all trees of fixed order containing no vertex of even degree are obtained for a>1 and α∈[0,1]; also, the maximum value of Rα0 in the mentioned class is determined for 0<α<1. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of SEIa and Rα0 are found for every real number α∉{0,1} and a>1. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of SEIa and Rα0 are determined when a>1 and α does not belong to the closed interval [0,1]; in the same class, the maximum values of Rα0 are also found for 0<α<1. The graphs that achieve the obtained extremal values are also determined.