Injective and Relative Injective Zagreb Indıces of 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.

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.

Journal of Mathematics

Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing subdivision-related operations S , Q , R , and T on G , the four new graphs S G (subdivision graph), Q G (edge-semitotal), R G (vertex-semitotal), and T G (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H , we define F -sum graphs (denoted by G + F H ) which are obtained by Cartesian product of F G and H , where F ∈ S , R , Q , T . In this study, we determine first general Zagreb co-index of graphs under operations in the form of Zagreb indices and co-indices of their basic graphs.

   , 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.

2020

Let G = (V, E) be a graph. The first Zagreb index and second Zagreb index of G are defined as v∈V d 2 (v) and uv∈E d(u)d(v), respectively. Using first and second Zagreb indices of graphs, we in this note present sufficient conditions for some Hamiltonian properties of graphs.

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

2012

In this study, we first find formulae for the first and second Zagreb indices and coindices of certain classical graph types including path, cycle, star and complete graphs. Secondly we give similar formulae for the first and second Zagreb coindices.

Proyecciones (Antofagasta), 2020

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.

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.