In this paper, two new topological **indices** based on **Zagreb** **indices** are proposed. The exact values of these new topological **indices** are calculated for some stan- dard **graphs** and for the firefly **graphs**. These new **indices** can be used to investi- gate the chemical properties for some chemical compound such as drugs, bridge molecular graph etc. For the future work, instead of defining these new topolog- ical **indices** based on the degrees of the vertices, we can redefine them based on the degrees of the edges by defining them on the line graph of any graph. Similar calculations can be computed to indicate different properties of the graph.

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The **first** and **second** **Zagreb** index of degree splitting of standard **graphs** and special **graphs** are studied in [11],[12]. My research is to find the **first** and **second** **Zagreb** index of degree splitting of molecular **graphs** like silicate networks, hexagonal network, honey comb network, Theorem 1

A topological index ,which is a graph invariant it does not depend on the labeling or pictorial representation of the graph, is a numerical parameter mathematically derived from the graph structure. The topological **indices** of molecular **graphs** are widely used for establishing correlations between the structure of a molecular compound and its physico- chemical properties or biological activity . These **indices** are used in quantitive structure property relations (QSPR) research. The **first** distance based topological index was pro- posed by Wiener in 1947 for modeling physical properties of alcanes [1], and after him, hundred topological **indices** were defined by chemists and mathematicians and so many properties of chemical structures were studied. More than forty years ago Gutman and Trinajsti´ c defined **Zagreb** **indices** which are degree based topological **indices** [2]. These topological **indices** were proposed to be measures of branching of the carbon-atom skele- ton in [3]. The **first** and **second** **Zagreb** **indices** of a simple connected graph G defined as follows;

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Throughout this paper graph means simple connected **graphs**. Let G be a connected graph with vertex and edge sets V(G) and E(G), respectively. As usual, the degree of a vertex u of G is denoted by deg(u) and it is defined as the number of edges incident with u. A topological index is a real number related to a graph. It must be a structural invariant, i.e., it preserves by every graph automorphisms. There are several topological **indices** have been defined and many of them have found applications as means to model chemical, pharmaceutical and other properties of molecules.

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Let G be simple graph. The first Zagreb index is the sum of squares of degree of vertices and second Zagreb index is the sum of the products of the degrees of pairs of[r]

ABSTRACT: In this paper, we investigate several topological **indices** in honeycomb **graphs**: Randić connectivity index, sum-connectivity index, atom-bond connectivity index, geometric-arithmetic index, **First** and **Second** **Zagreb** **indices** and **Zagreb** polynomials. Formulas for computing the above topological descriptors in honeycomb **graphs** are given.

Topological **indices** are the numerical values which are associated with a graph structure. These graph invariants are utilized for modeling information of molecules in structural chem- istry and biology. Over the years many topological **indices** are proposed and studied based on degree, distance and other parameters of graph. Some of them may be found in [5, 7]. Histori- cally **Zagreb** **indices** can be considered as the **first** degree-based topological **indices**, which came into picture during the study of total π-electron energy of alternant hydrocarbons by Gutman and Trinajsti c ´ in 1972 [9]. Since these **indices** were coined, various studies related to different aspects of these **indices** are reported, for detail see the papers [4, 6, 8, 12, 17] and the references therein.

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Let G = (V (G), E(G)) be graph with vertex set V (G) and edge set E(G). All the **graphs** considered in this paper are sim- ple and connected. Graph theory has successfully provided chemists with a variety of useful tools [1, 8, 10, 11], among which are the topological **indices**. In theoretical chemistry, assigning a numerical value to the molecular structure that will closely correlate with the physical quantities and activi- ties. Molecular structure descriptors (also called topological **indices**) are used for modeling physicochemical, pharmaco- logic, toxicologic, biological and other properties of chemical compounds. **Zagreb** **indices** were introduced more than forty years ago by Gutman and Trinajestic [9]. The **Zagreb** **indices** are found to have applications in QSPR and QSAR studies as well, see [3]. One of the recently introduced **indices** called Reverse **Zagreb** **indices** by Ediz [5] and he obtained the max- imum and minimum **graphs** with respect to the **first** reverse **Zagreb** alpha index and minimum **graphs** with respect to the **first** reverse **Zagreb** beta index and the **second** reverse Za- greb index. Kulli [15, 16] defined **first** and **second** reverse

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W G d u v The Wiener index is a classical distance based topological index. It was **first** introduced by H. Wiener [26] with an application to chemistry. The Wiener index has been extensively studied by many chemists and mathematicians. For more details about Wiener index the reader may refer to [5, 6, 12, 16, 20]. The **first** and **second** **Zagreb** **indices** of a graph denoted by M G 1 ( ) and M G 2 ( ) , respectively, are degree based topological **indices** introduced by Gutman and Trinajstić [15]. These two **indices** are defined as

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n e G n e G over all edges e uv of G , where n e G 1 ( | ) and n e G 2 ( | ) are respectively the number of vertices of G lying closer to vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to vertex u . In this paper, we determine the n-vertex bipartite unicyclic **graphs** with the **first**, the **second**, the third and the fourth smallest Szeged **indices**.

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In [11], Gutman showed thatamong all trees with n ≥ 5 vertices, the extremal (minimal and maximal) trees regarding the multiplicative **Zagreb** **indices** are the path and star . Eliasi [7] identified thirteen trees with the **first** through ninth greatest multiplicative **Zagreb** index among all trees of order n. In the same line, Eliasi and Ghalavand [10] introduced a graph transformation, which decreases . By applying this operation, they identified the eight classes of trees with the **first** through eighth smallest among all trees of order n ≥ 12. Also the effects on the **first** general **Zagreb** index were observed when some operations including edge moving, edge separating and edge switching were applied to the **graphs** [18]. Moreover, by using majorization theory, the authors [18] obtained the largest or smallest **first** general **Zagreb** **indices** among some classes of connected **graphs**. Some more outstanding mathematical studies on multiplicative **Zagreb** **indices** are [4, 8, 9, 19, 21, 24].

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Inspired by the chemical applications of higher-order connectivity index (or Randic′ index), we consider here the higher-order **first** **Zagreb** index of a molecular graph. In this paper, we study the linear regression analysis of the **second** order **first** **Zagreb** index with the entropy and acentric factor of an octane isomers. The linear model, based on the **second** order **first** **Zagreb** index, is better than models corresponding to the **first** **Zagreb** index and F-index. Further, we compute the **second** order **first** **Zagreb** index of line **graphs** of subdivision **graphs** of 2D- lattice, nanotube and nanotorus of [ , ], tadpole **graphs**, wheel **graphs** and ladder **graphs** .

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Abstract. The **first** **Zagreb** index of a graph G is the sum of squares of the vertex degrees in a graph and the **second** **Zagreb** index of G is the sum of products of degrees of adjacent vertices in G. The imbalance of an edge in G is the numerical difference of degrees of its end vertices and the irregularity of G is the sum of imbalances of all its edges. In this paper, we extend the concepts of these topological **indices** for signed **graphs** and discuss the corresponding results on signed **graphs**.

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[14] named the sum in (1) the third **Zagreb** index, and established new bounds on the **first** and **second** Zagrab **indices** that depend on irr(G). Zhou and Luo obtained the relationship between irregularity and **first** **Zagreb** index of **graphs**, and also they determined the **graphs** with maximum irregularity among trees and unicyclic **graphs** with given matching number and number of pendent vertices [19, 29]. Hansen and Melot determined the maximum irregularity of **graphs** with n vertices and m edges [17]. Moreover, Abdo and Dimitrov considered the irregularity of **graphs** under several graph operations [5]. Previously, we characterized all **graphs** with the **second** minimum of the irregularity in [20]. Also, we studied in [15, 21], trees and unicyclic **graphs** whose irregularity is extremal. More works about this graph invariant have been reported in [2, 9, 18, 22−24].

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E G e e e , thus possessing n vertices and m edges. The degree d v ( ) of the vertex v V G ( ) is the number of **first** neighbors of v. The edge of the graph G, connecting the vertices u and v, will be denoted by uv . Throughout this paper, the **graphs** considered are assumed to be connected.

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Let G be an overfull graph with maximum degree Δ. **First**, we classify all such **graphs** up to isomorphism, for 1 ≤ Δ ≤ 4. In Theorems 9 – 11, we construct these **graphs**. In the following Theorems, let G is an overfull graph. It should be noted that, according to Theorem 2 such **graphs** have no pendant ices Hence, if Δ = 1, then G is not overfull.

The subdivision graph S(G) of a graph G is the graph obtained from G by replacing each of its edges by a path of length 2, or equivalently by inserting an additional vertex into each edge of G. Subdivision **graphs** are used to obtain several mathematical and chemical properties of more complex **graphs** from more basic **graphs** and there are many results on these **graphs**. Similarly the r-subdivision graph of G denoted by S r (G) is defined by adding r vertices to each edge, [23], [25]. Then, we obtain the double **graphs** of these subdivision **graphs**. These subdivision and r-subdivision **graphs** were recently studied by several authors, [12, 18, 22, 23, 24, 25, 28, 29]. In that paper, ten types of **Zagreb** **indices** including **first** and **second** **Zagreb** **indices** and multiplicative **Zagreb** **indices** that we shall be concentrating in this paper on were calculated.

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The **Zagreb** **indices** are the oldest graph invariants used in mathematical chemistry to predict the chemical phenomena. In this paper we define the multiple versions of **Zagreb** **indices** based on degrees of vertices in a given graph and then we compute the **first** and **second** extremal **graphs** for them.

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These **indices** were defined in 1972 by Gutman and Trinajstic, [5]. In [3], some results on the **first** **Zagreb** index together with some other **indices** were obtained. In [4], the mul- tiplicative versions of these **indices** were studied. Some relations between **Zagreb** **indices** and some other **indices** such as ABC, GA and Randic **indices** were obtained in [7]. **Zagreb** **indices** of subdivision **graphs** were studied in [9] and these were calculated for the line **graphs** of the subdivision **graphs** in [8]. A more generalized version of subdivision **graphs** is called r-subdivision **graphs** and **Zagreb** **indices** of r-subdivision **graphs** were calculated in [10]. These **indices** were calculated for several important graph classes in [11]. In this paper, we deal with the edge-**Zagreb** matrices defined by means of the **second** **Zagreb** index, see e.g. [1] and [6].

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