On the Inverse Problem for Some Topological Indices

The study of the inverse problem (IP) based on the topological indices (TIs) deals with the numerical relations to TIs. Mathematically, the IP can be expressed as follows: given a graph parameter/TI that assigns a non-negative integer value g to every graph within a given family G of graphs, find some G ∈ G for which TI G = g . It was initiated by the Zefirov group in Moscow and later Gutman et al. proposed it. In this paper, we have established the IP only for the Y -index, Gourava indices, second hyper-Zagreb index, reformulated first Zagreb index, and reformulated F -index since they are closely related to each other. We have also studied the same which is true for the molecular, tree, unicyclic, and bicyclic graphs.

On the Reformulated Multiplicative First Zagreb Index of Trees and Unicyclic Graphs

The multiplicative first Zagreb index of a graph H is defined as the product of the squares of the degrees of vertices of H . The line graph of a graph H is denoted by L H and is defined as the graph whose vertex set is the edge set of H where two vertices of L H are adjacent if and only if they are adjacent in H . The multiplicative first Zagreb index of the line graph of a graph H is referred to as the reformulated multiplicative first Zagreb index of H . This paper gives characterization of the unique graph attaining the minimum or maximum value of the reformulated multiplicative first Zagreb index in the class of all (i) trees of a fixed order (ii) connected unicyclic graphs of a fixed order.

Modified Reduced Neighborhood Topological Indices of Cycloalkanes, Clycloalkenes, and Cycloalkynes

In this paper, some newly introduced topological indices, namely, modified reduced neighborhood first Zagreb index (〖RNM〗_1^* (G)) and modified forgotten reduced neighborhood index (〖RNM〗_3^* (G)) of a graph, G are used to establish the general formula for the molecular graphs of cycloalkanes, cycloalkenes, and cycloalkynes. We also calculate the exact values of some special types of these molecular graphs by attaching the alkyls instead of hydrogen atoms. Also, the QSPR analysis for these newly introduced indices is studied, and we exhibit a good correlation with characteristics on these molecular graphs.

Energy of Nonsingular Graphs: Improving Lower Bounds

Let G be a simple graph of order n and A be its adjacency matrix. Let λ 1 ≥ λ 2 ≥ … ≥ λ n be eigenvalues of matrix A . Then, the energy of a graph G is defined as ε G = ∑ i = 1 n λ i . In this paper, we will discuss the new lower bounds for the energy of nonsingular graphs in terms of degree sequence, 2-sequence, the first Zagreb index, and chromatic number. Moreover, we improve some previous well-known bounds for connected nonsingular graphs.

Nirmala energy

A novel vertex-degree-based topological invariant, called Nirmala index, was recently put forward, defined as the sum of the terms \(\sqrt{d(u)+d(v)}\) over all edges \(uv\) of the underlying graph, where \(d(u)\) is the degree of the vertex \(u\). Based on this index, we now introduce the respective ``Nirmala matrix'', and consider its spectrum and energy. An interesting finding is that some spectral properties of the Nirmala matrix, including its energy, are related to the first Zagreb index.

Topological Study of Zeolite Socony Mobil-5 via Degree-Based Topological Indices

Graph theory is a subdivision of discrete mathematics. In graph theory, a graph is made up of vertices connected through edges. Topological indices are numerical parameters or descriptors of graph. Topological index tells the symmetry of compound and helps us to compare those mathematical values, with boiling point, melting point, density, viscosity, hydrophobic surface area, polarity, etc., of that compound. In the present research paper, degree-based topological indices of Zeolite Socony Mobil-5 are calculated. Names of those topological indices are Randić index, first Zagreb index, general sum connectivity index, hyper-Zagreb index, geometric index, ABC index, etc.

Topological Indices of Some New Graph Operations and Their Possible Applications

A chemical graph theory is a fascinating branch of graph theory which has many applications related to chemistry. A topological index is a real number related to a graph, as its considered a structural invariant. It’s found that there is a strong correlation between the properties of chemical compounds and their topological indices. In this paper, we introduce some new graph operations for the first Zagreb index, second Zagreb index and forgotten index "F-index". Furthermore, it was found some possible applications on some new graph operations such as roperties of molecular graphs that resulted by alkanes or cyclic alkanes.

First Zagreb index on a fuzzy graph and its application

The Zagreb index (ZI) is a very important graph parameter and it is extensively used in molecular chemistry, spectral graph theory, network theory and several fields of mathematics and chemistry. In this article, the first ZI is studied for several fuzzy graphs like path, cycle, star, fuzzy subgraph, etc. and presented an ample number of results. Also, it is established that the complete fuzzy graph has maximal first ZI among n-vertex fuzzy graphs. Some bounds of first ZI are discussed for Cartesian product, composition, union and join of two fuzzy graphs. An algorithm has been designed to calculate the first ZI of a fuzzy graph. At the end of the article, a multi-criteria decision making (MCDM) method is provided using the first ZI of a fuzzy graph to find the best employee in a company. Also a comparison is provided among related indices on the result of application and shown that our method gives better results.