Beyond the Zagreb indices

The augmented eccentric connectivity index of a graph which is a generalization of eccentric connectivity index is defined as the summation of the quotients of the product of adjacent vertex degrees and eccentricity of the concerned vertex of a graph. In this paper we established some relationships between augmented eccentric connectivity index and several other graph invariants like number of vertices, number of edges, maximum vertex degree, minimum vertex degree, the total eccentricity index the first Zagreb indices and the second multiplicative Zagreb index.

Assigning numbers to graphs serves as a fundamental tool in graph theory and enables various analytical, computational, and visualization techniques for studying and understanding graph structures and properties. Edge Contraction is a process that separates edges from the graph while also joining the two previously linked vertices. In this paper, we study metric dimension and degree-based topological indices of Möbius Ladder and edge contracted Möbius Ladder like first Zagreb index, second Zagreb index, first Zagreb Hyper index, second Zagreb Hyper index, Augmented Zagreb index, Randic index, general Randic index, and Harmonic index.

The first and second Zagreb indices of a graph are equal‎, ‎respectively‎, ‎to‎ ‎the sum of squares of the vertex degrees‎, ‎and the sum of the products of‎ ‎the degrees of pairs of adjacent vertices‎. ‎We now consider analogous‎ ‎graph invariants‎, ‎based on the second degrees of vertices (number of‎ ‎their second neighbors)‎, ‎called leap Zagreb indices‎. ‎A number of their‎ ‎basic properties is established‎.

The Zagreb indices are the oldest among all degree-based topological indices. For a connected graph G, the first Zagreb index M1(G) is the sum of the term dG(u)+dG(v) corresponding to each edge uv in G, that is, M1 , where dG(u) is degree of the vertex u in G. In this chapter, the authors propose a weighted first Zagreb index and calculate its values for some standard graphs. Also, the authors study its correlations with various physico-chemical properties of octane isomers. It is found that this novel index has strong correlation with acentric factor and entropy of octane isomers as compared to other existing topological indices.

The first Zagreb index M1 (G) defined as the sum of squares of the degrees of the vertices. In this paper we compare and analyze numerous lower bounds for the first Zagreb index involving the number of vertices, the number of edges and the maximum and minimum vertex degree. In addition, we propose new lower bound and correct the equality case in [E.I. Milovanović and I.Ž. Milovanović, Sharp Bounds for the first Zagreb index and first Zagreb coindex, Miskolc Mathematical notes, 16 (2015) 1017-1024].

On topological properties of the line graphs of subdivision graphs of certain nanostructures

The Zagreb indices of graphs based on four new operations related to the lexicographic product

We introduce several series of novel ZE-isomerism descriptors derived directly from two-dimensional molecular topology. These descriptors make use of a quantity named ZE-isomerism correction, which is added to the vertex degrees of atoms connected by double bonds in Z and E configurations. This approach is similar to the one described previously for topological chirality descriptors (Golbraikh, A., et al. J. Chem. Inf. Comput. Sci. 2001, 41, 147-158). The ZE-isomerism descriptors include modified molecular connectivity indices, overall Zagreb indices, extended connectivity, overall connectivity, and topological charge indices. They can be either real or complex numbers. Mathematical properties of different subgroups of ZE-isomerism descriptors are discussed. These descriptors circumvent the inability of conventional topological indices to distinguish between Z and E isomers. The applicability of ZE-isomerism descriptors to QSAR analysis is demonstrated in the studies of a series of 131 anticancer agents inhibiting tubulin polymerization.

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Molecular descriptors also known as topological indices are numeric quantities which characterize its topology and some properties of molecular graph. The 2-distance degree of a vertex u in a graph G is equal to the number of its second neighbors and topological indices based on second neighbors of end vertices are named as leap indices. Leap indices have gained attention of many researchers. Therefore in this paper, we find some exact values for the leap Zagreb indices and leap hyper-indices of carbon Nanocones.

Numerous studies based on mathematical models and tools indicate that there is a strong inherent relationship between the chemical properties of the chemical compounds and drugs with their molecular structures. In the last two decades, the graph-theoretic techniques are frequently used to analyse the various physicochemical and structural properties of the molecular graphs which play a vital role in chemical engineering and pharmaceutical industry. In this paper, we compute Zagreb indices of the generalized sum graphs in the form of the different indices of their factor graphs, where generalized sum graphs are obtained under the operations of subdivision and strong product of graphs. Moreover, the obtained results are illustrated with the help of particular classes of graphs and analysed to find the efficient subclass with dominant indices.

We present Araneola, a scalable reliable application-level multicast system for highly dynamic wide-area environments. Araneola supports multi-point to multi-point reliable communication in a fully distributed manner while incurring constant load on each node. For a tunable parameter k /spl ges/ 3, Araneola constructs and dynamically maintains an overlay structure in which each node's degree is either k or k + 1, and roughly 90% of the nodes have degree k. Empirical evaluation shows that Araneola's overlay structure achieves three important mathematical properties of k-regular random graphs (i.e., random graphs in which each node has exactly k neighbors) with N nodes: (i) its diameter grows logarithmically with N; (ii) it is generally k-connected; and (iii) it remains highly connected following random removal of linear-size subsets of edges or nodes. The overlay is constructed at a very low cost: each join, leave, or failure is handled locally, and entails the sending of only about 3k messages in total. Given this overlay, Araneola disseminates multicast messages by gossiping over the overlay's links. We show that compared to a standard gossip-based multicast protocol, Araneola achieves substantial improvements in load, reliability, and latency. Finally, we present an extension to Araneola in which the basic overlay is enhanced with additional links chosen according to geographic proximity and available bandwidth. We show that this approach reduces the number of physical hops messages traverse without hurting the overlay's robustness.

Sufficient conditions for hamiltonian properties of graphs based on the difference of Zagreb indices

Zagreb eccentricity indices were proposed analogously to Zagreb indices already known and used for almost forty years. For a connected graph, the first Zagreb eccentricity index is defined as the sum of the squares of the eccentricities of the vertices, and the second Zagreb eccentricity index is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. We report mathematical prop- erties, especially lower and upper bounds of trees and general graphs in terms of graph invariants and the corresponding extremal graphs, Nordhaus-Gaddum-type results, and the ordering of trees with small and large Zagreb eccentricity indices. (doi: 10.5562/cca1801)

Recently the differences of Zagreb indices are reported by Furtula, Gutman and Ediz. The Zagreb coindices are defined as M1(G) = ∑uv ∉ E(G) [d(u) + d(v)] and M2(G) = ∑uv ∉ E(G)d(u)d(v), where d(u) is the degree of the vertex u of G. So far, the difference M1 and M2 has not been studied. We show that this difference is closely related to the vertex degree-based invariant RM2=∑uv ∉ E(G)(d(u)-1)(d(v)-1), and explore their basic mathematical properties and present explicit formulas for this new invariant under several graph operations.