Saira Hameed

In this modern era, graph theory has become an integral part of science and technology. It has enormous applications in handling various design-based problems. In this study, we present a new approach that increases the ability of graph theory to deal with uncertain challenges. The spherical fuzzy graph addresses the uncertainty domain more broadly, going beyond classical fuzzy graphs other than generalized fuzzy graph types. The concept of the spherical fuzzy planarity value provides a new method to evaluate the edge intersection and order in the graph. The ability of the spherical fuzzy planar graph (SFPG) to model complex relationships can enable more precise and reliable network designs and analyses. In this work, the concepts of spherical fuzzy multigraph (SFMG), spherical strong and weak edges, and the value of planarity for spherical fuzzy graphs (SFGs) are introduced. Moreover, the concept of the degree of planarity within the context of spherical fuzzy planar graphs and the notion of strong and weak faces are introduced. Additionally, we delve into the construction of spherical fuzzy dual graphs, which can be realized in cases where the fuzzy graph is planar or possesses a degree of planarity ≥0.67. This notion also serves as the foundation for certain basic theorems. We talk about some significant findings related to this subject. We discuss some major results linked to this topic. To show the worth and importance of our work, we also provide a real-world application.

The graph spectra analyze the structure of the graph using eigenspectra. The spectral graph theory deals with the investigation of graphs in terms of the eigenspectrum. In this paper, the sequence of lower bounds for the spectral radius of digraph $\mathcal{D}$ having at least one doubly adjacent vertex in terms of indegree is proposed. Particularly, it is exhibited that $\rho (\mathcal{D})\ge {\alpha }^j=\sqrt{{\displaystyle \frac{{\sum }_{p=1}^m{({\chi }_{j+1}^{(p)})}^2}{{\sum }_{p=1}^m{({\chi }_j^{(p)})}^2}}}$, such that equality is attained iff $\mathcal{D}=\overleftrightarrow{\mathfrak{G}}+$ {$\mathcal{DE}\notin$ Cycle}, where each component of associated graph is a k-regular or $(k_1,k_2)$ semiregular bipartite. By utilizing the sequence of lower bounds of the spectral radius of $\mathcal{D}$, the sequence of upper bounds of energy of $\mathcal{D}$, where the sequence decreases when $\sqrt{{\displaystyle \frac{\mathbf{e}}{\mathfrak{U}}}}\le {\alpha }^j$ and increases when $\sqrt{{\displaystyle \frac{\mathbf{e}}{\mathfrak{U}}}}>{\alpha }^j$, are also proposed. All of the obtained inequalities are elaborated using examples. We also discuss the monotonicity of these sequences.

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.

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.