In QSAR and QSPR studies the most commonly used topological index was proposed by chemist Milan Randić in 1975 called Randić branching index or path-one molecular connectivity index, 1χ and it has many applications. In the extension of connectivity indices, in early 1990s, chemist Milan Randic´ introduced variable Randić index defined as ∑v1v2∈E(G) ((dv1 + θ*)(dv2 + θ*))−1/2, where θ* is a non-negative real number and dv1 is the degree of Vertex V1 in G. The main objective of the present study is to prove the conjecture proposed in [19]. In this study, we will show that the Pn (path graph) has the maximum variable connectivity index among the collection of trees whose order is n, where n ≥ 4.

A $k$-CEC graph is a graph $G$ which has connected domination number $\gamma_{c}(G) = k$ and $\gamma_{c}(G + uv) < k$ for every $uv \in E(\overline{G})$. A $k$-CVC graph $G$ is a $2$-connected graph with  $\gamma_{c}(G) = k$ and $\gamma_{c}(G - v) < k$ for any $v \in V(G)$. A graph is said to be maximal $k$-CVC if it is both $k$-CEC and $k$-CVC. Let $\delta$, $\kappa$, and $\alpha$ be the minimum degree, connectivity, and independence number of $G$, respectively. In this work, we prove that for a maximal $3$-CVC graph, if $\alpha = \kappa$, then $\kappa = \delta$. We additionally consider the class of maximal $3$-CVC graphs with $\alpha < \kappa$ and $\kappa < \delta$, and prove that every $3$-connected maximal $3$-CVC graph when $\kappa < \delta$ is Hamiltonian connected.

Recently, some techniques such as adding whiskers and attaching graphs to vertices of a given graph, have been proposed for constructing a new Vertex decomposable graph. In this paper, we present a new method for constructing Vertex decomposable graphs. Then we use this construction to generalize the result due to Cook and Nagel.

Let G = (V; E) be a simple connected graph with n vertices, m edges and sequence of Vertex degrees d1  d2 


 dn > 0, di = d(vi), where vi 2 V. With i  j we denote adjacency of vertices vi and vj. The general sum{connectivity index of graph is de ned as   (G) = P  (di + dj ) i j, where  is an arbitrary real number. In this paper we determine relations between   + (G) and   + 1(G), where  and  are arbitrary real numbers, and obtain new bounds for   (G). Also, by the appropriate choice of parameters  and  , we obtain a number of old/new inequalities for di erent Vertex{degree{based topological indices.

The graph Γ(R1◦R2) of the lexicographic product of two commutative rings R1; R2 is considered. It was shown that Γ(R1 ◦ R2) is connectedand diam(Γ(R1 ◦ R2)) ≤ 2. We get the several expressions for finding theconnectivity κ(Γ(R1 ◦ R2)) when certain conditions are given

Let $G$ be a simple graph with Vertex set $V=V(G)$ and edge set $E=E(G)$. For a real function $f$ defined on nonnegative real numbers, the Vertex-degree function index $H_{f}(G)$ is defined as $$H_{f}(G)=\sum_{u\in V(G)}f(d_{u}).$$ In this paper we introduce the Vertex-degree function index $H_{f}(D)$ of a digraph $D$. After giving some examples and basic properties of $H_{f}(D)$, we find the extremal values of $H_{f}$ among all tournaments with a fixed number of vertices, when $f$ is a continuous and convex (or concave) real function on $\left[ 0,+\infty \right)$.

The Vertex version of PI index is a molecular structure descriptor which is similar to Vertex version of Szeged index. In this paper, we compute the Vertex-PI index of TUC4C8(S), TUC4C8(R) and HAC5C7[r, p].

Let G = (V (G), E(G)) be a simple, finite and undirected graph of order n. A k-Vertex weighting of a graph G is a mapping w: V (G) → {1, . . ., k}. A k-Vertex weighting induces an edge labeling fw: E(G) → N such that fw(uv) = w(u) + w(v). Such a labeling is called an edge-coloring k-Vertex weighting if fw(e) ̸ = fw(e′ ) for any two adjacent edges e and e′ . Denote by μ ′ (G) the minimum k for G to admit an edge-coloring k-Vertex weighting. In this paper, we determine μ ′ (G) for some classes of graphs.