On Trees with a Given Number of Vertices of Fixed Degree and Their Two Bond Incident Degree Indices - Abeer M. Albalahi; Muhammad Rizwan; Akhlaq A. Bhatti; Ivan Gutman; Akbar Ali; Tariq Alraqad; Hicham Saber (Axioms)
This paper is mainly concerned with the study of two bond incident degree (BID) indices, namely the variable sum exdeg index <semantics>SEIa</semantics> and the general zeroth-order Randić index <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics>. The minimum values of <semantics>SEIa</semantics> and <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics> in the class of all trees of fixed order containing no vertex of even degree are obtained for <semantics>a>1</semantics> and <semantics>α∈[0,1]</semantics>; also, the maximum value of <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics> in the mentioned class is determined for <semantics>0<α<1</semantics>. Moreover, in the family of all trees of fixed order and with a given number of vertices of even degrees, the extremum values of <semantics>SEIa</semantics> and <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics> are found for every real number <semantics>α∉{0,1}</semantics> and <semantics>a>1</semantics>. Furthermore, in the class of all trees of fixed order and with a given number of vertices of maximum degree, the minimum values of <semantics>SEIa</semantics> and <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics> are determined when <semantics>a>1</semantics> and <semantics>α</semantics> does not belong to the closed interval <semantics>[0,1]</semantics>; in the same class, the maximum values of <semantics><mmultiscripts>Rα<none></none><mprescripts></mprescripts><none></none>0</mmultiscripts></semantics> are also found for <semantics>0<α<1</semantics>. The graphs that achieve the obtained extremal values are also determined.