Researchers are still drawn to research the physical molecular and chemical structure of benzenoid hydrocarbons, unsaturated, fully conjugated compounds with hexagonal arrangements that exhibit remarkable features in relation to aromaticity. For chemical graphs in many dimensions, structures, or networks, topological indices or numerical descriptors have been employed for decades to link key physicochemical parameters with crucial molecular structural features including melting, boiling point, enthalpy, and cyclicity. For this work, the inverse degrees of the molecular or chemical structure or graphs being studied are used to calculate the reverse-degree-based topological indices. In molecular graph theory, reverse-degree-based topological descriptors are a relatively new method for analyzing chemical networks and structures. In this study, we suggest a reverse-degree-based topological representation. We computed particular types of descriptors of two-dimensional (2-D) coronene fractal formations with a variety of reverse-degree-based topological indices, such as the reverse-degree-based topological index of the first, second, and hyper Zagreb, forgotten, geometric arithmetic, atom-bond-connectivity, and the Randic index.

Let $\Lambda$ denote a commutative ring with unity and $\mathfrak{D}\left(\Lambda \right)$ denote a collection of all annihilating ideals from $\Lambda$. An annihilator intersection graph of $\Lambda$ is represented by the notation $\mathcal{AIG}\left(\Lambda \right)$. This graph is not directed in nature, where the vertex set is represented by $\mathfrak{D}{\left(\Lambda \right)}^{*}$. There is a connection in the form of an edge between two distinct vertices $\varsigma$ and $\varrho$ in $\mathcal{AIG}\left(\Lambda \right)$ iff $Ann\left(\varsigma \varrho \right)\ne Ann\left(\varsigma \right)\cap Ann\left(\varrho \right)$. In this work, we begin by categorizing commutative rings $\Lambda$, which are finite in structure, so that $\mathcal{AIG}\left(\Lambda \right)$ forms a star graph/2-outerplanar graph, and we identify the inner vertex number of $\mathcal{AIG}\left(\Lambda \right)$. In addition, a classification of the finite rings where the genus of $\mathcal{AIG}\left(\Lambda \right)$ is 2, meaning $\mathcal{AIG}\left(\Lambda \right)$ is a double-toroidal graph, is also investigated. Further, we determine $\Lambda$, having a crosscap 1 of $\mathcal{AIG}\left(\Lambda \right),$ indicating that $\mathcal{AIG}\left(\Lambda \right)$ is a projective plane. Finally, we examine the domination number for the annihilator intersection graph and demonstrate that it is at maximum, two.

${\mathfrak{R}}_{\mathfrak{c}}$ represents commutative rings that have unity elements. The collection of all zero-divisor elements in ${\mathfrak{R}}_{\mathfrak{c}}$ are represented by $D\left({\mathfrak{R}}_{\mathfrak{c}}\right)$. We denote an extended zero-divisor graph by the notation ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ of ${\mathfrak{R}}_{\mathfrak{c}}.$ This graph has a set of vertices in $D{\left({\mathfrak{R}}_{\mathfrak{c}}\right)}^{*}$. The graph ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ illustrates interactions among the zero-divisor elements of ${\mathfrak{R}}_{\mathfrak{c}}.$ Specifically, two different vertices u and y are connected in ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ iff $u{\mathfrak{R}}_{\mathfrak{c}}\cap Ann\left(y\right)$ is non-null or $y{\mathfrak{R}}_{\mathfrak{c}}\cap Ann\left(u\right)$ is non-null. The main idea for this work is to systematically analyze the ring ${\mathfrak{R}}_{\mathfrak{c}}$ which is finite for the unique aspect of their extended zero-divisor graph. This study particularly focuses on instances where the extended zero-divisor graph has a genus or crosscap of two. Furthermore, this work aims to thoroughly characterize finite ring ${\mathfrak{R}}_{\mathfrak{c}}$ wherein the extended zero-divisor graph ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ has an outerplanarity index of two. Finally, we determine the book thickness of ${\daleth }^{\prime }\left({\mathfrak{R}}_{\mathfrak{c}}\right)$ for genus at most one.

The zero-divisor graph of a commutative ring $\mathfrak{R}$ with a nonzero identity, denoted by $\Gamma \left(\mathfrak{R}\right)$, is an undirected graph where the vertex set $Z{\left(\mathfrak{R}\right)}^{*}$ consists of all nonzero zero-divisors of $\mathfrak{R}$. Two distinct vertices a and b in $\Gamma \left(\mathfrak{R}\right)$ are adjacent if and only if $ab=0$. The normalized Laplacian spectrum of zero-divisor graphs has been studied extensively due to its algebraic and combinatorial significance. Notably, Pirzada and his co-authors computed the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for specific values of $\mathfrak{n}$ in the set $\{pq,p^{2}q,p^3,p^4\}$, where p and q are distinct primes satisfying $p<q$. Motivated by their work, this article investigates the normalized Laplacian spectrum of $\Gamma \left({\mathbb{Z}}_{\mathfrak{n}}\right)$ for a more general class of $\mathfrak{n}$, where $\mathfrak{n}$ is represented as $p_1^{T_1}{p}_2^{T_2}$, with $p_1$ and $p_2$ being distinct primes $(p_1<p_2)$, and $T_1,T_2$ are positive integers.

Chemical graph theory, a branch of computational and applied mathematics, covers a very wide range of topics. As a result, the world of applied sciences heavily relies on graph theory. The concept of partition dimension has significant importance in the field of chemical graph theory. Although certain graphs have bounded partition dimensions, a graph’s partition dimension may be constant. In this study, we look at two alternative chemical structures made of an octagonal grid: nanosheets and nanotubes. We determined the partition dimension of an octagonal grid-generated nanosheet to be 3, and the partition dimension of a nanotube to be limited from 4.