An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

Topological Indices and f-Polynomials on Some Graph Products

We obtain inequalities involving many topological indices in classical graph products by using the f-polynomial. In particular, we work with lexicographic product, Cartesian sum and Cartesian product, and with first Zagreb, forgotten, inverse degree and sum lordeg indices.

On sufficient conditions for a graph to be \(k\)-path-coverable, \(k\)-edge-hamiltonian, Hamilton-connected, traceable and \(k^{-}\)-independent

The inverse degree of a graph was defined as the sum of the inverses of the degrees of the vertices. In this paper, we focus on finding sufficient conditions in terms of the inverse degree for a graph to be \(k\)-path-coverable, \(k\)-edge-hamiltonian, Hamilton-connected and traceable, respectively. The results obtained are not dropped.

Some Bounds on Zeroth-Order General Randić Index

For a graph G without isolated vertices, the inverse degree of a graph G is defined as I D ( G ) = ∑ u ∈ V ( G ) d ( u ) − 1 where d ( u ) is the number of vertices adjacent to the vertex u in G. By replacing − 1 by any non-zero real number we obtain zeroth-order general Randić index, i.e., 0 R γ ( G ) = ∑ u ∈ V ( G ) d ( u ) γ , where γ ∈ R − { 0 } . Xu et al. investigated some lower and upper bounds on I D for a connected graph G in terms of connectivity, chromatic number, number of cut edges, and clique number. In this paper, we extend their results and investigate if the same results hold for γ < 0 . The corresponding extremal graphs have also been identified.

On the Inverse Degree Polynomial

Using the symmetry property of the inverse degree index, in this paper, we obtain several mathematical relations of the inverse degree polynomial, and we show that some properties of graphs, such as the cardinality of the set of vertices and edges, or the cyclomatic number, can be deduced from their inverse degree polynomials.

f-Polynomial on Some Graph Operations

Given any function f : Z + → R + , let us define the f-index I f ( G ) = ∑ u ∈ V ( G ) f ( d u ) and the f-polynomial P f ( G , x ) = ∑ u ∈ V ( G ) x 1 / f ( d u ) − 1 , for x > 0 . In addition, we define P f ( G , 0 ) = lim x → 0 + P f ( G , x ) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others.

More on inverse degree and topological indices of graphs

The inverse degree of a graph G with no isolated vertices is defined as the sum of reciprocal of vertex degrees of the graph G. In this paper, we obtain several lower and upper bounds on inverse degree ID(G). Moreover, using computational results, we prove our upper bound is strong and has the smallest deviation from the inverse degree ID(G). Next, we compare inverse degree ID(G) with topological indices (Randic index R(G), geometric-arithmetic index GA(G)) for chemical trees and also we determine the n-vertex chemical trees with the minimum, the second and the third minimum, as well as the second and the third maximum of ID - R. In addition, we correct the second and third minimum Randic index chemical trees in [16].

A short note on hyper Zagreb index

In this paper, we present and analyze the upper and lower bounds on the Hyper Zagreb index $\chi^2(G)$ of graph $G$ in terms of the number of vertices $(n)$, number of edges $(m)$, maximum degree $(\Delta)$, minimum degree $(\delta)$ and the inverse degree $(ID(G))$. In addition, we give a counter example on the upper bound of the second Zagreb index for Theorems 2.2 and 2.4 from \cite{ranjini}. Finally, we present lower and upper bounds on $\chi^2(G)+\chi^2(\overline G)$, where $\overline G$ denotes the complement of $G$.