Open Journal of Discrete Applied Mathematics (ODAM)

An edge irregular \(k\)-labeling of a graph \(G\) is a labeling of vertices of \(G\) with labels from the set \(\{1,2,3,\dots,k\}\) such that no two edges of \(G\) have same weight. The least value of \(k\) for which a graph \(G\) has an edge irregular \(k\)-labeling is called the edge irregularity strength of \(G\). Ahmad et. al. [1] showed the edge irregularity strength of some particular classes of Toeplitz graphs. In this paper we generalize those results and finds the exact values of the edge irregularity strength for some generalize classes of Toeplitz graphs.

The eccentric atom-bond sum-connectivity \(\left(ABSC_{e}\right)\) index of a graph \(G\) is defined as \(ABSC_{e}(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e_{u}+e_{v}-2}{e_{u}+e_{v}}}\), where \(e_{u}\) and \(e_{v}\) represent the eccentricities of \(u\) and \(v\) respectively. This work presents precise upper and lower bounds for the \(ABSC_{e}\) index of graphs based on their order, size, diameter, and radius. Moreover, we find the maximum and minimum \(ABSC_{e}\) index of trees based on the specified matching number and the number of pendent vertices.

Let \(G = (V(G), E(G))\) be a graph with minimum degree at least \(1\). The inverse degree of \(G\), denoted \(Id(G)\), is defined as the sum of the reciprocals of degrees of all vertices in \(G\). In this note, we present inverse degree conditions for Hamiltonian and traceable graphs.

This note presents some upper bounds for the size of the upper deg-centric grapg \(G_{ud}\) of a simple connected graph G. Amongst others, a result for graphs for which a compliant graph \(G\) has \(G_{ud} \cong \overline G\) is presented. Finally, results for size minimality in respect upper deg-centrication and minimum size of such graph \(G\) are presented.

There are three different kinds of topological indices: spectrum-based, degree-based, and distance-based. We presented the \(K\)-swapped network for \(t\)-regular graphs in this study. We also computed various degree-based topological indices of the \(K\)-swapped network for \(t\)-regular graphs, eye, and \(n\)-dimensional twisted cube network. The metrics used to analyze the abstract structural characteristics of networks are called topological indices. We also calculate each of the aforementioned networks M-polynomials. A graph can be used to depict an interconnection network’s structure. The processing nodes in the network are represented by vertices, while the links connecting the processor nodes are represented by edges. We can quickly determine the diameter and degree between the nodes based on the graph’s tpology. A key component of graph theory are graph invariants, which identify the structural characteristics of networks and graphs. Furthermore doescribed by graph invariants are computer, social, and internet networks.

Shadi I.K et al. [1] introduced the edge hub number of graphs. This work extends the concept to fuzzy graphs. We derive several properties of edge hub number of fuzzy graphs and establish some relations that connect the new parameter with other fuzzy graph parameters. Also, some bounds of such a parameter are investigated. Moreover, we provide empirical evidence examples to elucidate the behavior and implications of edge hub number of fuzzy graph parameters.

In this paper, we give a relationship between the covering number of a simple graph \(G\), \(\beta(G)\), and a new parameter associated to \(G\), which is called 2-degree-packing number of \(G\), \(\nu_2(G)\). We prove that \[\lceil \nu_{2}(G)/2\rceil\leq\beta(G)\leq\nu_2(G)-1,\] for any simple graph \(G\), with \(|E(G)|>\nu_2(G)\). Also, we give a characterization of connected graphs that attain the equalities.

Let \(A\) and \(B\) be two graph and \(P(A,z)\) and \(P(B,z)\) are their chromatic polynomial, respectively. The two graphs \(A\) and \(B\) are said to be chromatic equivalent denoted by \( A \sim B \) if \(P(A,z)=P(B,z)\). A graph \(A\) is said to be chromatically unique(or simply \(\chi\)- unique) if for any graph \(B\) such that \(A\sim B \), we have \(A\cong B\), that is \(A\) is isomorphic to \(B\). In this paper, the chromatic uniqueness of a new family of \(6\)-bridge graph \(\theta(r,r,s,s,t,u)\) where \(2\leq r\leq s \leq t\leq u\) is investigated.

In mathematical chemistry, a large number of topological indices are used to predict the physicochemical properties of compounds, especially in the study of quantitative structure-proerty relationship (QSPR).
However, many topological indices have almost the same predictive ability. In this paper, we focus on how to use fewer topological indices to predict the physicochemical properties of compounds through the QSPR analysis of connectivity indices of benzene hydrocarbons.

In this paper, we obtain the bounds for the Laplacian eigenvalues of a weighted graph using traces. Then, we find the bounds for the Kirchhoff and Laplacian Estrada indices of a weighted graph. Finally, we define the Laplacian energy of a weighted graph and get the upper bound for this energy.