A vertex u of a graph G = ( V , E ), ve -dominates every edge incident to u , as well as every edge adjacent to these incident edges. A set S ⊆ V is a vertex-edge dominating set (or a ved-set for short) if every edge of E is ve- dominated by at least one vertex of S . The vertex-edge domination number is the minimum cardinality of a ved-set in G . In this paper, we investigate the graphs having unique minimum ved-sets that we will call UVED-graphs. We start by giving some basic properties of UVED-graphs. For the class of trees, we establish two equivalent conditions characterizing UVED-trees which we subsequently complete by providing a constructive characterization.

Let [Formula: see text] be a simple connected graph. The first Zagreb index denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and edge-vertex domination number for trees and characterized the extremal trees attaining the bound.

Let [Formula: see text] be a simple connected graph. The modified first Zagreb index is denoted by [Formula: see text] is defined as [Formula: see text]. We present a lower bound for [Formula: see text] in terms of order and total domination number for trees and characterize the extremal trees attaining the bound.

Let [Formula: see text] be vertices of a graph [Formula: see text] with degree of the vertices being [Formula: see text] and [Formula: see text] respectively. First, let us define the weight of the edge [Formula: see text] as twice the value of [Formula: see text] in [Formula: see text]. Let us define [Formula: see text], the harmonic index of the graph [Formula: see text], as the sum obtained by adding the weight assigned to every edge of [Formula: see text]. In this paper, for the class of trees, we shall obtain an upper bound for the harmonic index [Formula: see text] in terms of the edge-vertex domination number and the order of [Formula: see text]. Also, we shall ascertain that the equality is true by characterizing the collection of all extremal trees attaining this bound.

A vertex [Formula: see text] of a graph [Formula: see text] is said to vertex-edge dominate every edge incident to [Formula: see text], as well as every edge adjacent to these incident edges. A subset [Formula: see text] is a vertex-edge dominating set (ve-dominating set) if every edge of [Formula: see text] is vertex-edge dominated by at least one vertex of [Formula: see text]. A vertex-edge dominating set is said to be total if its induced subgraph has no isolated vertices. The minimum cardinality of a total vertex-edge dominating set of [Formula: see text], denoted by [Formula: see text], is called the total vertex-edge domination number of [Formula: see text]. In this paper, we prove that for every nontrivial tree of order [Formula: see text], with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize extremal trees attaining the lower bound.

Let G be a graph having the set of edges EG. Represent by dGu the degree of a vertex u of G. The Sombor (SO) index of G is defined as SOG=∑uv∈EGdGu2+dGv2. The length of a shortest cycle in a graph G is known as the girth of G. A connected graph with the same order and size is usually referred to as a (connected) unicyclic graph. This paper reports the characterization of the graphs possessing the first-two maximum values of the SO index in the class of all (connected) unicyclic graphs with a fixed girth and order.

Let [Formula: see text] be a simple graph. A set [Formula: see text] is called a super dominating set if for every vertex [Formula: see text], there exist [Formula: see text] such that [Formula: see text]. The minimum cardinality of a super dominating set of [Formula: see text], denoted by [Formula: see text], is called the super domination number of graph [Formula: see text]. Characterization of trees with [Formula: see text] is presented.

For a graph [Formula: see text] with vertex set [Formula: see text] and edge set [Formula: see text], a subset [Formula: see text] of [Formula: see text] is the total edge dominating set if every edge in [Formula: see text] is adjacent to at least one edge in [Formula: see text]. The minimum cardinality of a total edge dominated set, denoted by [Formula: see text], is called the total edge domination number of a graph [Formula: see text]. We prove that for every tree [Formula: see text] of diameter at least two with [Formula: see text] leaves and [Formula: see text] support vertices we have [Formula: see text], and we characterize the trees attaining each of the bounds.

Let \(G=(V,E)\) be a simple graph. A set \(S\subseteq V\) is a dominating set if every vertex in \(V \setminus S\) is adjacent to a vertex in \(S\). The domination number of a graph \(G\), denoted by \(\gamma(G)\) is the minimum cardinality of a dominating set of \(G\). A set \(D \subseteq E\) is an edge dominating set if every edge in \(E\setminus D\) is adjacent to an edge in \(D\). The edge domination number of a graph \(G\), denoted by \(\gamma'(G)\) is the minimum cardinality of an edge dominating set of \(G\). We characterize trees with domination number equal to twice edge domination number.

This thesis work on the two parameters that is very important domination parameters, one parameter is known as domination number and other parameter is called as total domination number. S is defined as a set of vertices in a graph G. We characterize a set S of vertices in a graph G with no segregated vertices to be a semitotal overwhelming arrangement of G in the event that it is a ruling arrangement of G and furthermore every vertex in S is inside separation 2 of another vertex of S. The semitotal domination number, indicated by is the base cardinality of a semitotal ruling arrangement of G. We demonstrate that on the off chance that G is an associated graph on n ≥ 4 vertices, at that point and we describe the trees and diagrams of least degree 2 arriving at this bound. __________________________________________________*****_________________________________________________