A vertex w ∈ V H distinguishes (or resolves) two elements (edges or vertices) a , z ∈ V H ∪ E H if d w , a ≠ d w , z . A set W m of vertices in a nontrivial connected graph H is said to be a mixed resolving set for H if every two different elements (edges and vertices) of H are distinguished by at least one vertex of W m . The mixed resolving set with minimum cardinality in H is called the mixed metric dimension (vertex-edge resolvability) of H and denoted by m dim H . The aim of this research is to determine the mixed metric dimension of some wheel graph subdivisions. We specifically analyze and compare the mixed metric, edge metric, and metric dimensions of the graphs obtained after the wheel graphs’ spoke, cycle, and barycentric subdivisions. We also prove that the mixed resolving sets for some of these graphs are independent.

The concept of Fermatean fuzzy set offers a more comprehensive perspective on Pythagorean fuzzy sets, with greater potential for practical applications due to its broader scope compared to Pythagorean fuzzy sets. In contrast, distance measures serve as effective information measures in decision-making using deep learning methods. This research paper introduces a novel distance measure within the Fermatean fuzzy framework. This new distance measure exhibit enhanced performance indices compared to existing distance measures discussed in previous literature. The newly proposed distance operators based on Fermatean fuzzy sets offer a fresh approach by considering all three membership functions of Fermatean fuzzy sets, which differs from existing practices. This distance operator also incorporate all membership functions to prevent errors caused by exclusions, which have been observed in other distance operators. Several theorems are provided to validate the alignment of this new Fermatean fuzzy distance technique with properties expected of distance operators. The comparative analysis of the newly introduced Fermatean fuzzy distance measure with similar existing distance techniques is conducted through numerical comparison and also in relation to their applications, aiming to highlight the superiority of the proposed Pythagorean fuzzy distance techniques over the existing ones. The application of the proposed Fermatean fuzzy distance measure is demonstrated in scenarios involving pattern classification and bidirectional approximate reasoning.

A molecular (chemical) graph is a simple connected graph, where the vertices represent the compound’s atoms and the edges represent bonds between the atoms, and the degree (valence) of every vertex (atom) is not more than four. In this paper, we determine the edge metric basis and edge metric dimension (EMD) of the complex molecular graph of a one-heptagonal carbon nanocone ( $HCN_{7}(q)$ ). We prove that only three non-adjacent vertices are the minimum requirement for the identification of all the edges in $HCN_{7}(q)$ , uniquely.

For an undirected connected graph G = G(V, E) with vertex set V(G) and edge set E(G), a subset R of V is said to be a resolving in G, if each pair of vertices (say a and b; a ≠ b) in G satisfy the relation d(a, k) ≠ d(b, k), for at least one member k in R. The minimum set R with this resolving property is said to be a metric basis for G, and the cardinality of such set R, is referred to as the metric dimension of G, denoted by dim _v (G). In this manuscript, we consider a complex molecular graph of one-heptagonal carbon nanocone (represented by HCN _s ) and investigate its metric basis as well as metric dimension. We prove that just three specifically chosen vertices are enough to resolve the molecular graph of HCN _s . Moreover, several theoretical as well as applicative properties including comparison have also been incorporated.

Let $G=G(V,E)$ be a non-trivial simple connected graph. The length of the shortest path between two vertices $p$ and $q$, represented by $d(p, q)$, is called the distance between the vertices $p$ and $q$. The distance between an edge $\varepsilon=pq$ and a vertex $r$ in $G$ is defined as $d(\varepsilon, r)=\mbox{min}\{d(p, r), d(q, r)\}$. If $d(r, p)\neq d(r, q)$, then the vertex $r$ is said to distinguish (resolve or recognize) two elements (edges or vertices) $p, q\in V\cup E$. The minimum cardinality of a subset $R$ ($R_e$) of vertices such that all other vertices (edges) of the graph $G$ are uniquely determined by their distances to the vertices in $R$ ($R_e$) is the metric dimension (edge metric dimension) of a graph $G$. In this article, we consider a family of pentagonal circular ladder $(P_m)$ and investigate its edge metric dimension. We show that, for $P_m$ the edge metric dimension is strictly greater than its metric dimension. Additionally, we answer a problem raised in the recent past, regarding the edge metric dimension of a family of a planar graph $R_m$ (exists in the literature).