Iranian Journal of Mathematical Chemistry
Year:2024 | Volume:15 | Issue:2|Papers Count:6

‎Partition dimension problems involve dividing a graph's vertex set into a minimum number of disjoint sets so that each vertex is different with respect to the representation from each disjoint set‎. ‎As a result of the development of this method‎, ‎a number of applications have arisen in a number of fields such as drug design‎, ‎navigation of robots‎, ‎pattern recognition‎, ‎and image processing‎. ‎In this paper‎, ‎we have calculated the partition dimension of oxide and {zigzag benzenoid networks‎, ‎and the subdivision of benzenoid hydrocarbon and triangular benzene networks.

‎The Laplacian eigenvalues and polynomials of the networks play an essential role in understanding the relations between the topology and the dynamic of networks‎. ‎Generally‎, ‎computation of the Laplacian spectrum of a network is a hard problem and there are just a few classes of graphs with the property that their spectra have been completely computed‎. ‎Laplacian spectrum for $ n$-prism networks was investigated in [Liu et al.‎, ‎Neurocomputing 198 (2016) 69-73]‎. ‎In this paper‎, ‎we give a method for calculating the eigenvalues and characteristic polynomial of the Laplacian matrix of a generalized $n$-prism network‎. ‎We show how such large networks can be constructed from small graphs by using graph products‎. ‎Moreover‎, ‎our results are used to obtain the Kirchhoff index and the number of the spanning trees in the generalized $n$-prism networks‎. ‎We also give some examples of applications‎, ‎that explain the usefulness and efficiency of the proposed method‎.

‎Let $\mathcal{R}$ be the commutative ring $\mathcal{R}=\mathbb{Z}_{p^2}[x]/\langle x^{2} \rangle$ with identity and ${Z^{*}}(\mathcal{R})$ be the set of all non-zero zero-divisors of $\mathcal{R}$‎. ‎Then‎, ‎$\Gamma(\mathcal{R})$ is said to be a zero-divisor graph if and only if $a \cdot b= 0$ where $a,b \in V(\Gamma(\mathcal{R})) = {Z^{*}}(\mathcal{R})$ and $(a,b) \in E(\Gamma(\mathcal{R}))$‎. ‎Let $\lambda_1,\lambda_2,\dots,\lambda_n$ be the eigenvalues of the adjacency matrix‎, ‎and let $\mu_1,\mu_2,\dots,\mu_n$ be the eigenvalues of the Laplacian matrix of $\Gamma(\mathcal{R})$‎. ‎Then %the energy of $\Gamma(\mathcal{R})$ is defined as the sum of the absolute values of the eigenvalues of the graph $\Gamma(\mathcal{R})$ and the Laplacian energy of $\Gamma(\mathcal{R})$ is the sum of the absolute deviations of its Laplacian matrix's eigenvalues of the graph $\Gamma(\mathcal{R})$‎. ‎In this paper‎,‎we discuss the energy $\mathcal{E}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\lambda_{i}}$ and the Laplacian energy $\mathcal{LE}(\Gamma(\mathcal{R}))=\sum_{i=1}^n \abs{\mu_{i}-\frac{2m}{n}}$ where $n$ and $m$ are the order and size of $\Gamma(\mathcal{R})$‎.

‎In this paper‎, ‎we delve into a coefficient inverse problem linked to the bioheat equation‎, ‎a pivotal component in medical research concerning phenomena such as temperature response and blood perfusion during surface heating‎. ‎By considering factors like heat transfer between tissue and blood in capillaries and incorporating the geometric intricacies of the skin‎, ‎we confine our analysis to a one-dimensional domain‎. ‎Our approach involves transforming the original problem into one concerning the reconstruction of a multiplicative source term within a parabolic equation‎. ‎Subsequently‎, ‎we utilize integral conditions to derive a specific integro-differential equation‎, ‎accompanied by the requisite initial and boundary conditions‎. ‎Leveraging a spectral method‎, ‎we streamline the modified problem into a linear system of algebraic equations‎. ‎To accomplish this‎, ‎we employ appropriate regularization algorithms to obtain stable approximations for the derivatives of perturbed boundary data and to effectively solve the resultant system of equations‎.

In this paper‎, ‎algorithms are used to compute distance-based topological indices for the Complete Binary Tree (CBT) and the Complete Ternary Tree (CTT)‎. ‎Computation of distance-based topological indices is complex for varied heights of CBT and CTT‎. ‎Hence designed algorithms to compute distance between any-to-any vertex made this possible to compute the required topological indices for CBT and CTT‎. ‎The distance calculator algorithm designed for this study can also be customized in digital chemical structures‎, ‎mathematical chemistry‎, ‎network traffic control in wireless networks‎, ‎search applications‎, ‎high bandwidth routing‎, ‎parse construction in compilers‎, ‎and memory management.

‎For any graph $G$‎, ‎the modified Sombor index is defined as the reciprocal of the well-known Sombor Index‎. ‎The girth of $G$‎, ‎by short $g(G)$‎, ‎is the length of the smallest cycle in $G$‎. ‎A graph with exactly one cycle is a unicyclic graph‎. ‎If it is further‎, ‎connected‎, ‎it is a connected unicyclic graph‎. ‎In this article‎, ‎we achieved the modified Sombor index for a collection of graphs‎, ‎connected unicyclic graphs and also obtained the maximum modified Sombor index for the class of unicyclic graphs based on the restrictions by a fixed girth‎.