DEGREE DISTANCE AND GUTMAN INDEX OF INCREASING TREES

Abstract

‎‎The Gutman index and degree distance of a connected graph $G$ are defined as‎ ‎begin{eqnarray*}‎ ‎textrm{Gut}(G)=sum_{{u,v}subseteq V(G)}d(u)d(v)d_G(u,v)‎, ‎end{eqnarray*}‎ ‎and‎ ‎begin{eqnarray*}‎ ‎DD(G)=sum_{{u,v}subseteq V(G)}(d(u)+d(v))d_G(u,v)‎, ‎end{eqnarray*}‎ ‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_G(u,v)$ is the distance between vertices $u$ and $v$‎. ‎In this paper‎, ‎through a recurrence equation for the Wiener index‎, ‎we study the first two‎ ‎moments of the Gutman index and degree distance of increasing‎ ‎trees‎.