Let G = (V, E) be a connected graph and d(u, v) denote the distance between the vertices u and v in G. A set of vertices W resolves a graph G if every vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G and is denoted by dim(G). Let J_2n,m be a m-level gear graph obtained by m-level wheel graph W_2n,m ≅ mC_2n + k₁ by alternatively deleting n spokes of each copy of C_2n and J_3n be a generalized gear graph obtained by alternately deleting 2n spokes of the wheel graph W_3n. In this paper, the metric dimension of certain gear graphs J_2n,m and J_3n generated by wheel has been computed. Also this study extends the previous result given by Tomescu et al. in 2007.

This paper investigates the metric dimensions of the polygonal networks, particularly, the subdivided honeycomb network, Aztec diamond as well as the subdivided Aztec diamond network. A polygon is any two-dimensional shape formed by straight lines. Triangles, quadrilaterals, pentagons, and hexagons are all representations of polygons. For instance, hexagons help us in many models to construct honeycomb network, where n is the number of hexagons from a central point to the borderline of the network. A subdivided honeycomb network is obtained by adding additional vertices on each edge of . An Aztec diamond network of order is a lattice comprises of unit squares with center satisfying . The subdivided Aztec diamond network is obtained by adding additional vertices to each edge of . In this work, our main aim is to establish the results to show that the metric dimensions of and are 2 and 3 for and , respectively. In the end, some open problems are listed with regard to metric dimensions for -subdivisions of and .

A relatively new and rapidly expanding area of mathematics research is the study of graphs’ spectral properties. Spectral graph theory plays a very important role in understanding certifiable applications such as cryptography, combinatorial design, and coding theory. Nonassociative algebras, loop groups, and quasigroups are the generalizations of associative algebra. Many studies have focused on the spectral properties of simple graphs connected to associative algebras like finite groups and rings, but the same research direction remains unexplored for loop groups and quasigroups. Eigenvalue analysis, subgraph counting, matrix representation, and the combinatorial approach are key techniques and methods in our work. The main purpose of this paper is to characterize finite quasigroups with the help of relatively prime graphs. Moreover, we investigate the structural and spectral properties of these graphs associated with finite quasigroups in the forms of star graphs, eigenvalues, connectivity, girth, clique, and chromatic number.

Let $$\Delta$$ be a simplicial complex on the vertex set $$V$$. For $$m=1,2,3,\dots$$, the notion of $$m$$-th $$\mathcal{NF}$$-complex of $$\Delta$$, $$\delta^{(m)}_{\mathcal{NF}}(\Delta)$$, was introduced by Hibi and Mahmood in [5], where $$\delta^{(m)}_{\mathcal{NF}}(\Delta)=\delta_{\mathcal{NF}}(\delta^{(m-1)}_{\mathcal{NF}}(\Delta))$$ with setting $$\delta^{(1)}_{\mathcal{NF}}(\Delta)=\delta_{\mathcal{NF}}(\Delta)$$ such that $$\delta_{\mathcal{NF}}(\Delta)$$ is the Stanley–Reisner complex of the facet ideal of $$\Delta$$. The $$\mathcal{NF}$$-number of $$\Delta$$ is the least positive integer $$q$$ for which $$\delta^{(q)}_{\mathcal{NF}}(\Delta)\simeq\Delta$$. In this paper, we investigated the $$\mathcal{NF}$$-number of two copies of complete graphs $$K_n$$ joined by one common vertex $$\{u\}$$. At the end, we also provided an explicit example for the case of two copies of $$K_5$$ joined by common vertex $$\{u\}$$.