Let $G=(V,E)$, $V={v_1,v_2,ldots,v_n}$, be a simple graph with$n$ vertices, $m$ edges and a sequence of vertex degrees$Delta=d_1ge d_2ge cdots ge d_n=delta$, $d_i=d(v_i)$. Ifvertices $v_i$ and $v_j$ are adjacent in $G$, it is denoted as $isim j$, otherwise, we write $insim j$. The first Zagreb index isvertex-degree-based graph invariant defined as$M_1(G)=sum_{i=1}^nd_i^2$, whereas the first Zag...

Given a tree T = (V,E), the second Zagreb index of T is denoted by M2(T ) = ∑ uv∈E d(u)d(v) and the Wiener polarity index of T is equal to WP (T ) = ∑ uv∈E(d(u)−1)(d(v)−1). Let π = (d1, d2, ..., dn) and π′ = (d1, d2, ..., dn) be two different non-increasing tree degree sequences. We write π π′, if and only if ∑n i=1 di = ∑n i=1 d ′ i, and ∑j i=1 di ≤ ∑j i=1 d ′ i for all j = 1, 2, ..., n. Let Γ...

In this paper we study lower bounds in a unified way for large family of topological indices, including the first variable Zagreb index M 1 ? . Our aim is to obtain sharp inequalities and characterize corresponding extremal graphs. The main results provide several vertex-degree-based indices. These are new even Zagreb, inverse forgotten

A topological index of a graph is a numeric quantity related to a structure of a molecule which is invariant under graph automorphism. Recently, Ghorbani and Hosseinzadeh introduced Fourth Zagreb index of graphs. In this paper we determine a closed formula of this new topological index of the famous Benzenoid family named Circumcoronene series of Benzenoid Hk.