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 $G$ 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.

The Indu-Bala product of graphs G and H consists of two disjoint copies of the join of G and H such that there is an  adjacency between the corresponding vertices in the two copies of H. A vertex subset S of a graph G = (V, E) is said to be a geodetic set if every vertex in G is in some u−v geodesic, where u and v are any two vertices in S. The minimum cardinality of such a set is the geodetic number of G. The vertex subset D of a graph G is said to be a dominating set if every vertex in G is either in D or adjacent to at least one vertex in D. The minimum cardinality of such a set is the domination number of G. In this work, the authors studied various geodetic and dominating extensions with respect to the Indu-Bala product of graphs. The Aα matrix associated with a graph is a convex linear combination of its adjacency matrix and degree diagonal matrix, offering deeper insights into the properties of both matrices. In this article the authors discuss the Aα spectrum of Indu-Bala product of graphs.

Let x be a vertex of a connected graph G and W  V (G) such that x =2 W. Then W is called an x-Steiner set of G if W [fxg is a Steiner set of G. The minimum cardinality of an x-Steiner set of G is defined as x-Steiner number of G and denoted by sx(G). Some general properties satisfied by these concepts are studied. The x-Steiner numbers of certain classes of graphs are determined. Connected graphs of order p with x-Steiner number 1 or p-1 are characterized. It is shown that for every pair a, b of integers with 2  a  b, there exists a connected graph G such that s(G) = a and sx(G) = b for some vertex x in G, where s(G) is the Steiner number of a graph.