The q-rung orthopair fuzzy set (q-ROFS), an extension of intuitionistic and Pythagorean fuzzy sets, offers greater flexibility in representing vague information with two possible outcomes, yes or no. The fuzzy Zagreb index is an important graph parameter, widely used in fields such as network theory, spectral graph theory, mathematics, and molecular chemistry. In this paper, the first and second Zagreb indices for q-rung orthopair fuzzy graphs (q-ROFGs) are introduced, and bounds for these indices are established, including their behavior in regular q-ROFGs. Additionally, it is explored, how various graph operations such as union, Cartesian product, direct product, and lexicographical product affect the first Zagreb index. Furthermore, a new approach is presented that combines Multiple-Attribute Decision-Making (MADM) with graph-based models to improve decision-making, particularly in vaccine selection. The methodology constructs a bipartite graph for each attribute, where virologists assign membership and non-membership values to vaccines. The Zagreb index is used to measure the importance of each vaccine, and a weighted aggregation technique normalizes the scores. The final ranking is derived from a computed score function. The results demonstrate the effectiveness of the approach in providing a systematic and mathematically rigorous framework for multi-attribute decision-making, with rank correlation analysis confirming its robustness compared to existing methods such as q-ROF PROMETHEE, q-ROF VICOR, q-ROF TOPSIS, q-ROFWG, and q-ROFWA.

Graph theory serves as an engaging arena for the investigation of proof methods within the field of discrete mathematics, and its findings find practical utility in numerous scientific domains. Chemical graph theory is a specialized branch of mathematics that uses graphs to represent and analyze the structure and properties of chemical compounds. Topological indices are mathematical properties of graphs that play a crucial role in chemistry. They provide a unique way to connect the structural characteristics of chemical compounds to their corresponding molecular graphs. The flabellum graph $F_n^{(k,j)}$ is obtained with the help of $k\ge 2$ duplicates of the cycle graph $C_n$ with a common vertex (known as, central vertex). Then, in j of these duplicates, additional edges are added, joining the central vertex to all non-adjacent vertices. In this article, we compute different degree-based topological indices for flabellum graphs, including some well known indices, such as the Randić index, the atom bond connectivity index, the geometric–arithmetic index, and the Zagreb indices. This research provides an in-depth examination of these specific indices within the context of flabellum graphs. Moreover, the behavior of these indices is shown graphically, in terms of the parameters $j,k$, and n. Additionally, we have extended the concept of the first Zagreb index, to address the issue of cybercrime. This application enables us to identify criminals who exhibit higher levels of activity and engagement in multiple criminal activities when compared to their counterparts. Furthermore, we conducted a comprehensive comparative analysis of the first Zagreb index against the closeness centrality measure. This analysis sheds light on the effectiveness and relevance of the topological index in the context of cybercrime detection and network analysis.

Let $G=(V,E)$ be a simple connected graph with vertex set V and edge set E, respectively. The term “anti-reciprocal eigenvalue property“ refers to a non-singular graph G for which, $-\frac{1}{\lambda }\in \sigma \left(G\right)$, whenever $\lambda \in \sigma \left(G\right)$, $\forall \lambda \in \sigma \left(G\right)$. Here, $\sigma \left(G\right)$ is the multiset of all eigenvalues of $A\left(G\right)$. Moreover, if multiplicities of eigenvalues and their negative reciprocals are equal, then that graph is said to have strong anti-reciprocal eigenvalue properties, and the graph is referred to as a strong anti-reciprocal graph (or $(-SR)$ graph). In this article, a new family of graphs $F_n^{(k,j)}$ is introduced and the energy of $F_5^{(k,\frac{k}{2})}$$k\ge 2$ is calculated. Furthermore, with the help of $F_5^{(k,\frac{k}{2})}$, some families of $(-SR)$ graphs are constructed.

Spectral graph theory is like a special tool for understanding graphs. It helps us find patterns and connections in complex networks, using the magic of eigenvalues. Let $\mathsf{G}$ be the graph and $\mathsf{A}(\mathsf{G})$ be its adjacency matrix, then $\mathsf{G}$ is singular if the determinant of the adjacency matrix $\mathsf{A}(\mathsf{G})$ is 0, otherwise it is nonsingular. Within the realm of nonsingular graphs, there is the concept of property $\mathtt{R}$, where each eigenvalue’s reciprocal is also an eigenvalue of $\mathsf{G}$. By introducing multiplicity constraints on both eigenvalues and their reciprocals, it becomes property $\mathtt{SR}$. Similarly, the world of nonsingular graphs reveals property $-\mathtt{R}$, where the negative reciprocal of each eigenvalue also finds a place within the spectrum of $\mathsf{G}$. Moreover, when the multiplicity of each eigenvalue and its negative reciprocal is equal, this results in a graph with a property of $-\mathtt{SR}$. Some classes of unweighted nonbipartite graphs are already constructed in the literature with the help of the complete graph ${\mathtt{K}}_n$ and a copy of the path graph ${\mathtt{P}}_4$ satisfying property $\mathtt{R}$ but not $\mathtt{SR}$. This article takes this a step further. The main aim is to construct several weighted classes of graphs which satisfy property $\mathtt{R}$ but not $\mathtt{SR}$. For this purpose, the weight functions are determined that enable these nonbipartite graph classes to satisfy the $-\mathtt{SR}$ and $\mathtt{R}$ properties, even if the unweighted graph does not satisfy these properties. Some examples are presented to support the investigated results. These examples explain how certain weight functions make these special types of graphs meet the properties $\mathtt{R}$ or $-\mathtt{SR}$, even when the original graphs without weights do not meet these properties.