Sadia Akhter

Many properties of the structure and dynamics of complex networks derive from the characteristics of the spectrum of the associated Laplacian matrix, specifically from the set of its eigenvalues. In this paper, we show that there exist graphs for which the ratio between the length of the spectrum (that is, the difference between the largest and smallest eigenvalues of the Laplacian matrix) and its spread (the difference between the second smallest eigenvalue and the smallest one) is equal to the golden ratio. We call such graphs Golden Laplacian Graphs (GLG). In this paper, we first find all such graphs with a number of nodes $n\le 10$. We then prove several graph-theoretic and algebraic properties that characterize these graphs. These graphs prove to be extremely robust, as they have large vertex and edge connectivity along with a large isoperimetric constant. Finally, we study the synchronization properties of GLGs, showing that they are among the top synchronizable graphs of the same size. Therefore, GLGs represent very good candidates for engineering and communication networks.

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.