The distance $d(v_a,v_b)$ between two vertices of a simple connected graph G is the length of the shortest path between $v_a$ and $v_b$. Vertices $v_a,v_b$ of G are considered to be resolved by a vertex v if $d(v_a,v)\ne d(v_b,v)$. An ordered set $W=\{v_1,v_2,v_3,\dots ,v_s\}\subseteq V\left(G\right)$ is said to be a resolving set for G, if for any $v_a,v_b\in V\left(G\right),$∃$v_i\in W\ni d(v_a,v_i)\ne d(v_b,v_i)$. The representation of vertex v with respect to W is denoted by $r\left(v\right|W)$ and is an s-vector(s-tuple) $(d(v,v_1),d(v,v_2),d(v,v_3),\dots ,d(v,v_s))$. Using representation $r\left(v\right|W)$, we can say that W is a resolving set if, for any two vertices $v_a,v_b\in V\left(G\right)$, we have $r\left(v_a\right|W)\ne r\left(v_b\right|W)$. A minimal resolving set is termed a metric basis for G. The cardinality of the metric basis set is called the metric dimension of G, represented by $dim\left(G\right)$. In this article, we study the metric dimension of two types of bicyclic graphs. The obtained results prove that they have constant metric dimension.

Intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and q-rung orthopair fuzzy sets (q-ROFS) are among those concepts which are widely used in real-world applications. However, these theories have their own limitations in terms of membership and non-membership functions, as they cannot be obtained from the whole unit plane. To overcome these restrictions, we developed the concept of a complex linear Diophantine fuzzy set (CLDFS) by generalizing the notion of a linear Diophantine fuzzy set (LDFS). This concept can be applied to real-world decision-making problems involving complex uncertain information. The main motivation behind this paper is to study the applications of CLDFS in a non-associative algebraic structure (AG-groupoid), which has received less attention as compared to associative structures. We characterize a strongly regular AG-groupoid in terms of newly developed CLDF-score left (right) ideals and CLDF-score $(0,2)$-ideals. Finally, we construct a novel approach to decision-making problems based on the proposed CLDF-score ideals, and some practical examples from civil engineering are considered to demonstrate the flexibility and clarity of the initiated CLDF-score ideals.

The distance between two vertices of a simple connected graph G, denoted as $d(u,v)$, is the length of the shortest path from u to v and is always symmetrical. An ordered subset $W=\left\{w_1,w_2,w_3,\cdots ,w_k\right\}$ of $V\left(G\right)$ is a resolving set for G, if for ∀$u,v\in V\left(G\right)$, there exists $w_i\in W$ ∋ $d(u,w_i)\ne d(v,w_i)$. A resolving set with minimal cardinality is called the metric basis. The metric dimension of G is the cardinality of metric basis of G and is denoted as $dim\left(G\right)$. For the graph $G_1=(V_1,E_{1,})$ and $G_2=(V_2,E_2)$, their join is denoted by $G_1+G_2$. The vertex set of $G_1+G_2$ is $V_1\cup V_2$ and the edge set is $E=E_1\cup E_2\cup \left\{uv,u\in V_1,v\in V_2\right\}$. In this article, we show that the metric dimension of the join of two path graphs is unbounded because of its dependence on the size of the paths. We also provide a general formula to determine this metric dimension. We also develop algorithms to obtain metric dimensions and a metric basis for the join of path graphs, with respect to its symmetries.

Characterizing simple connected graphs of order $ n $ having metric dimension $ n-2 $, solved by Chartrand et al. [Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99-113], is a foundational result in the study of metric dimension. This article presents a refined proof for the non-bipartite case of the original theorem. While this work does not present a new characterization result, its primary contribution is methodological: We reframe the original's lengthy case-by-case elimination argument as a series of standalone lemmas, which we use to formally establish the structural properties that such a graph must satisfy. Building upon these properties, we then provide a direct, constructive proof demonstrating that the graph structure is necessarily the join of a complete and empty graph. This method offers a more elegant argument for this important characterization and also provides a clearer understanding of why this specific graph family emerges.