On lower bounds for VDB topological indices of graphs

Abstract

Let $G =(V,E)$, $V=\{1,2,\ldots ,n\}$, be a simple graph of order $n$ and size $m$,  without isolated vertices. Denote by  $d_1\ge d_2 \ge \cdots \ge d_n$, $d_i=(d_i)$,  a sequence of its vertex degrees  in nondecreasing order. If vertices $i$ and $j$ are adjacent, we write $i\sim j$. Denote with $TI=TI(G)=\sum_{i\sim j}F(d_i,d_j)$ a class of vertex-degree-based invariants, where $F(x,y)$ may be any function satisfying the condition $F(x,y)=F(y,x)$. We define three new adjacency matrices for $G$ which are joined to $TI$, and then determine lower bounds for $TI$.