On the Occasion of his 80th Birthday Harmonic Index and Zagreb Indices of Vertex-Semitotal Graphs

Graph theory is one of the rising areas in mathematics due to its applications in many areas of science. Amongst several study areas in graph theory, spectral graph theory and topological descriptors are in front rows. These descriptors are widely used in QSPR/QSAR studies in mathematical chemistry. Vertex-semitotal graphs are one of the derived graph classes which are useful in calculating several physico-chemical properties of molecular structures by means of molecular graphs modelling the molecules. In this paper, several topological descriptors of vertexsemitotal graphs are calculated. Some new relations on these values are obtained by means of a recently defined graph invariant called omega invariant.


Introduction
Several topological graph indices have been defined and studied by many mathematicians and chemists. They are defined as topological graph invariants measuring several physical, chemical, pharmacological, pharmaceutical, biological, etc. properties of graphs which are modelling real life situations. They can be grouped mainly into three classes according to the way they are defined: by vertex degrees, by matrices or by distances. We consider degree based-topological indices of some derived graphs through this paper.
Let G = (V, E) be a simple graph with | V (G) |= n vertices and | E(G) |= m edges, where V (G) = {v 1 , v 2 , · · · , v n } and E(G) = {v i v j : v i , v j ∈ V (G)}. That is, we do not allow loops or multiple edges. For any vertex v ∈ V (G), we denote the degree of v by d G (v) or d v .
If v i and v j are adjacent vertices of G, and if the edge e connects them, this situation will be denoted by e = v i v j . In such a case, the vertices v i and v j are called adjacent vertices and the edge e is said to be incident to v i and v j . Adjacency and incidency play a very important role in the spectral graph theory, the sub area of graph theory dealing with linear algebraic study of graphs. The smallest and biggest vertex degrees in a graph will be denoted by δ and ∆, respectively.
Written with multiplicities, a degree sequence in general is written as Let D be a set of some non-decreasing non-negative integers. We say that a graph G is a realization of the set D if the degree sequence of G is equal to D.
, · · · , ∆ (a ∆ ) } be a realizable degree sequence and G be one of its realizations. The Ω(G) of G is defined in terms of the degree sequence as A vertex-semitotal graph T 1 (G) is constructed from G by inserting a new vertex for each edge of G and then by joining every inserted vertex to the end vertices of the corresponding edge, that is, by replacing each edge by a triangle. See Two of the most important topological graph indices are called the first and second Zagreb indices denoted by M 1 (G) and M 2 (G), respectively: They were first defined in 1972 by Gutman and Trinajstic, [12], and are referred to due to their uses in QSAR and QSPR studies. In [4], some results on the first Zagreb index together with some other indices are given. For some graph operations, these indices are calculated in [5,14,17].
The F -index or forgotten index of a graph G denoted by F (G) or M 3 (G) is defined as (2) Figure 1 The vertex-semitotal graph T 1 (G) of T 3,2 It was first appeared in the study of structure-dependency of total π-electron energy in 1972, [12]. Recently, this sum was named as the forgotten index or the F -index by Furtula and Gutman, [10].
The hyper-Zagreb index was defined as a variety of the classical Zagreb indices as see e.g. [10].
Inspired by the study of heat of formation for heptanes and octanes, in [9] Furtula et al. proposed an index called Augmented Zagreb index which gives a better prediction power. It is defined by The Harmonic index was introduced by Zhong [19] who found that it correlates well with Π-electron energy of benzenoid hydrocarbons and defined as Ranjini et al., [16], introduced the re-defined Zagreb indices, i.e. the redefined first, second and third Zagreb indices for a graph G and these are defined as Milicevic et al., [15], reformulated the Zagreb indices in terms of the edge degrees instead of the vertex-degrees as Aram and Dehgardi, [1], introduced the concept of reformulated F-index as Eliasi et. al. [8] introduced the multiplicative sum Zagreb index of G which is denoted by * 1 (G) and defined by * Xu et. al. [18] introduced the total multiplicative sum Zagreb index of a graph G denoted by T (G) and defined by Topological indices of some derived graphs, as subdivision, total, semitotal, line, paraline graphs are studied in [2] and [13]. In this paper, we examine some degree-based topological indices of vertex-semitotal graph which also is one of the derived graphs, and find relations between these topological indices.

Main Results
We first recall some results on the topological indices of the vertex-semitotal graphs: and

Proof. By Eqn. (3), we have
and the result follows.
Using Eqns. (7) and (8), we get the results for (ii) and (iii) by similar methods.
Theorem 7. Let G be a graph with order n = n(G) and size m = m(G). Reformulated forgotten index of T 1 (G) is Proof. For vertex-semitotal graph T 1 (G) of a graph G, there are two types of vertices: Firstly, the vertices corresponding to the vertices of G, secondly, the vertices corresponding to the edges of G. We will denote them v i and c ij , respectively. Depending on the nature of end vertices, we can divide the edges of T 1 into two types: i) v i v j -edge: an edge between two vertices in G. ii) v i c ij -edge: an edge between the vertices of G and the vertices corresponding to the edges of G. Using Eqn. (9), we have For v i c ij -edges in the second term, it is clear that every v i vertex of T 1 (G) is connected with d G (v i ) c ij vertices, each of degree 2. Therefore, corresponding to every vertex v i in Theorem 8. Let G be a graph of order n = n(G) and size m = m(G). Multiplicative sum and the result follows.
Theorem 9. Let G be a graph of order n = n(G) and size m = m(G). Total multiplicative sum Zagreb index of T 1 (G) is Proof. From the definition of T (G), we have (1 + d G (v i )) m(G) · 4 ( m(G) 2 ) and the result follows.

Conclusions
In this paper, we obtained the formulae for the topological indices, especially the Zagreb indices and harmonic index, of some class of derived graphs called vertex-semitotal graphs. We used for the first time a newly introduced graph invariant called omega invariant to obtain some of the relations. The methods used here can be applied to all topological graph indices and to other derived graphs and graph operations.