Some formulae for the Zagreb indices of graphs

In this paper, we introduce Zagreb Indices of Some New Graphs. Exactly, first index, second index and forgotten index. New graphs are generated from the initial graphs by graph operations. We also created some possible applications on the Zagreb indices as special cases.

Mathematics

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

Opuscula Mathematica, 2016

In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph.

   , where d G (v) is the degree of the vertex v. In this paper we compute these indices for link and splice of graphs. In continuation, with use these graph operations, we compute the first and the second multiplicative Zagreb indices for a class of dendrimers.

Analele Universitatii "Ovidius" Constanta - Seria Matematica, 2016

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as

International Journal of Combinatorics, 2012

The first and second Zagreb indices were first introduced by Gutman and Trinajstić (1972). It is reported that these indices are useful in the study of anti-inflammatory activities of certain chemical instances, and in elsewhere. Recently, the first and second Zagreb coindices, a new pair of invariants, were introduced in Došlić (2008). In this paper we introduce the and ()-analogs of the above Zagreb indices and coindices and investigate the relationship between the enhanced versions to get a unified theory.

Mathematics, 2015

The reformulated Zagreb indices of a graph are obtained from the classical Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of the end vertices of the edge minus 2. In this paper, we study the behavior of the reformulated first Zagreb index and apply our results to different chemically interesting molecular graphs and nano-structures.

Hacettepe Journal of Mathematics and Statistics, 2019

Recently, Furtula et al. [B. Furtula, I. Gutman, S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 2014] introduced a new vertex-degree-based graph invariant "reduced second Zagreb index" in chemical graph theory. Here we generalize the reduced second Zagreb index (call "general reduced second Zagreb index"), denoted by GRM α (G) and is defined as: , where α is any real number and d G (v) is the degree of the vertex v of G. Let G k n be the set of connected graphs of order n with k cut edges. In this paper, we study some properties of GRM α (G) for connected graphs G. Moreover, we obtain the sharp upper bounds on GRM α (G) in G k n for α ≥ -1/2 and characterize the extremal graphs.

Hacettepe Journal of Mathematics and Statistics, 2012

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. It is well-known that for connected or disconnected graphs, M2/m ≥ M1/n does not hold always. In K. C. Das (On comparing Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 63, 433–440, 2010), it has been shown that the above relation holds for a special kind of graph. Here we continue our search for special kinds of graph for which the above relation holds.

International Journal of Apllied Mathematics, 2020

Let G = (V, E) be a graph with n vertices and m edges. The hyper Zagreb index of G, denoted by HM (G), is defined as HM (G) = uv∈E(G) [d G (u) + d G (v)] 2 , where d G (v) denotes the degree of a vertex v in G. In this paper we compute the hyper Zagreb index of certain generalized graph structures such as generalized thorn graphs and generalized theta graphs. Also,for the first time, we determine exact values for hyper Zagreb index of some cycle related graphs, namely cycle with parallel P k chords, cycle with parallel C k chords and shell type graphs.