On Inequalities Involving Some Renowned Degree-Based Graph Invariants

Abstract

Let $G$ be a graph with the vertex set $V=\{v_1,\ldots,v_n\}$. We use the notation $i\sim j$ (respectively, $i\nsim j$) for indicating that the vertices $v_i$ and $v_j$ are adjacent (respectively, non-adjacent). Denote by $d_k$ the degree of the vertex $v_k$.
Most of the well--known vertex--degree--based topological indices and coindices can be represented in the forms
$
TI(G)=\sum_{i\sim j}F(d_i,d_j)
$
and
$
\overline{TI}(G)=\sum_{i\sim j}F(d_i,d_j),
$
respectively,
where $F$ is a real symmetric function depending on $d_i$ and $d_j$.
In this paper, several novel inequalities between some well-known topological indices/coindices are presented.
The graphs for which the obtained inequalities become equalities are also characterized.