In this paper, two new topological indices based on Zagrebindices 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.
The first and secondZagreb index of degree splitting of standard graphs and special graphs are studied in ,. My research is to find the first and secondZagreb 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 , 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 Zagrebindices which are degree based topological indices . These topological indices were proposed to be measures of branching of the carbon-atom skele- ton in . The first and secondZagrebindices of a simple connected graph G defined as follows;
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.
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 SecondZagrebindices 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 Zagrebindices 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 . 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.
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. Zagrebindices were introduced more than forty years ago by Gutman and Trinajestic . The Zagrebindices are found to have applications in QSPR and QSAR studies as well, see . One of the recently introduced indices called Reverse Zagrebindices by Ediz  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
W G d u v The Wiener index is a classical distance based topological index. It was first introduced by H. Wiener  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 secondZagrebindices of a graph denoted by M G 1 ( ) and M G 2 ( ) , respectively, are degree based topological indices introduced by Gutman and Trinajstić . These two indices are defined as
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.
In , Gutman showed thatamong all trees with n ≥ 5 vertices, the extremal (minimal and maximal) trees regarding the multiplicative Zagrebindices are the path and star . Eliasi  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  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 . Moreover, by using majorization theory, the authors  obtained the largest or smallest first general Zagrebindices among some classes of connected graphs. Some more outstanding mathematical studies on multiplicative Zagrebindices are [4, 8, 9, 19, 21, 24].
Inspired by the chemical applications of higher-order connectivity index (or Randic′ index), we consider here the higher-order firstZagreb index of a molecular graph. In this paper, we study the linear regression analysis of the second order firstZagreb index with the entropy and acentric factor of an octane isomers. The linear model, based on the second order firstZagreb index, is better than models corresponding to the firstZagreb index and F-index. Further, we compute the second order firstZagreb index of line graphs of subdivision graphs of 2D- lattice, nanotube and nanotorus of [ , ], tadpole graphs, wheel graphs and ladder graphs .
Abstract. The firstZagreb index of a graph G is the sum of squares of the vertex degrees in a graph and the secondZagreb 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.
 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 firstZagreb 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 . Moreover, Abdo and Dimitrov considered the irregularity of graphs under several graph operations . Previously, we characterized all graphs with the second minimum of the irregularity in . 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].
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.
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, , . 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 Zagrebindices including first and secondZagrebindices and multiplicative Zagrebindices that we shall be concentrating in this paper on were calculated.
The Zagrebindices are the oldest graph invariants used in mathematical chemistry to predict the chemical phenomena. In this paper we define the multiple versions of Zagrebindices based on degrees of vertices in a given graph and then we compute the first and second extremal graphs for them.
second multiplicative Zagreb coindices of G, respectively. In this article, we compute the first, second and third Zagrebindices and the first and second multiplicative Zagrebindices of some classes of dendrimers. The first and secondZagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.Also, the multiplicative Zagrebindices are computed using link of graphs.
These indices were defined in 1972 by Gutman and Trinajstic, . In , some results on the firstZagreb index together with some other indices were obtained. In , the mul- tiplicative versions of these indices were studied. Some relations between Zagrebindices and some other indices such as ABC, GA and Randic indices were obtained in . Zagrebindices of subdivision graphs were studied in  and these were calculated for the line graphs of the subdivision graphs in . A more generalized version of subdivision graphs is called r-subdivision graphs and Zagrebindices of r-subdivision graphs were calculated in . These indices were calculated for several important graph classes in . In this paper, we deal with the edge-Zagreb matrices defined by means of the secondZagreb index, see e.g.  and .