Modern network architectures, such as wireless sensor networks and cluster-based communication systems, demand a seamless integration of robust local connectivity with reliable global interconnectivity. In this paper, we model such layered networks via the Cartesian product F-m square K-n. The friendship graph F-m, formed by joining m triangles at a common vertex, captures the localized, hub-and-spoke clustering with enhanced local interconnectivity, beyond what a star graph can offer, and the complete graph K-n represents a uniformly interconnected backbone linking these clusters. In this setting, landmarks (or resolving sets) are crucial as they uniquely identify every node based solely on its distances to a carefully chosen subset of reference nodes, thereby facilitating efficient routing, localization, and fault detection. We establish exact values for the metric dimension of F-m square K-n for any parameter regime, thereby providing direct insights into the optimal selection of resolving sets for such architectures. Specifically, we prove that if n-1 >= 2m, then dim(F-m square K-n) = n-1, and if m/2 >= n-1, then dim(F-m square K-n) = m. For the intermediate case, m/2 < n -1 <2m, we develop a refined analysis by first deriving structural properties of metric bases and then formulating an optimization problem whose solution yields the desired metric dimension. In particular, for the intermediate regime m/2 < n -1 <2m, with (m, n) not equal (3, 4), we obtain the exact value dim(F-m square K-n) = inverted right perpendicular2(m+n-1)/3inverted left perpendicular All our proofs are constructive across all parameter regimes. Moreover, we give linear-time constructions of metric bases in all regimes and a backbone-assisted distributed rollout that uses O(n) backbone messages and O(|W|) local notifications. These results support layered/clustered network designs, including wireless sensor networks (WSNs) and backbone topologies, and provide explicit formulas and minimal-landmark constructions, together with corresponding bounds on control traffic and per-node state, for clustered backbone architectures modeled by F-m square K-n.

The metric dimension of a graph is the minimum number of landmark vertices required so that every vertex can be uniquely identified by its distances to the landmarks. This parameter captures the fundamental tradeoff between compact information encoding and unambiguous identification in networked systems. In this work, we determine exact value for the metric dimension of the Cartesian product K_1,m□ K_1,n, also known as hub-and-spoke grids, across all values of m and n. In addition, we present a constructive linear-time algorithm that builds a minimum resolving set, providing both theoretical guarantees and practical feasibility. We complement our results with visualization of parameter regimes that illustrate the design space. The findings establish design rules for minimizing landmark sensors and support applications in graph-based localization, monitoring networks, and intelligent information systems. Our results extend the theory of metric dimension and contribute efficient methods of direct relevance to information science and computational graph theory.

A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by psi(G), is the minimum cardinality of a doubly resolving set for G. In this paper, using adjacency resolving sets and dominating sets of graphs, we study doubly resolving sets in the corona product of graphs G and H, G circle dot H. First, we obtain the upper and lower bounds for the doubly resolving number of the corona product G circle dot H in terms of the order of G and the adjacency dimension of H, then we present several conditions that make each of these bounds feasible for the doubly resolving number of G circle dot H. Also, for some important families of graphs, we obtain the exact value of the doubly resolving number of the corona product.

Two vertices u,v in a connected graph G are doubly resolved by vertices x,y of G if d(v,x) - d(u,x) not equal d(v,y) - d(u,y). A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by psi(G), is the minimum cardinality of a doubly resolving set for G. The aim of this paper is to investigate doubly resolving sets in the lexicographic product graphs. It is proved that if H is not an element of{P-3,P-3(sic)} or G does not have any vertex of degree 1, then psi(G[H]) =dim(G[H]). Also psi(G[H]) is computed in other cases.

For an ordered set W = {w1, w2, ... , wk} of vertices and a vertex v in a connected graph G, the k-vector r(v|W) = (d(v, w1), d(v, w2), ... , d(v, wk)) is called the metric representation of v with respect to W, where d(x, y) is the distance between the vertices x and y. A set W is called a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension dim(G), and a resolving set of minimum cardinality is a basis of G. The corona product, G 0 H of graphs G and H is obtained by taking one copy of G and n(G) copies of H, and by joining each vertex of the ith copy of H to the ith vertex of G. In this paper, we obtain bounds for dim(G 0 K1), characterize all graphs G with dim(G 0 K1) = dim(G), and prove that dim(G 0 K1) = n - 1 if and only if G is the complete graph Kn or the star graph K1,n-1.

Two vertices u,v in a connected graph G are doubly resolved by x,y∈G if d(v,x)−d(u,x)≠d(v,y)−d(u,y).A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W. Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for the graph G. In this paper all graphs G with ψ(G)=2 are characterized by using 2-connected subgraphs of G.

‎For an ordered set $W=\{w_1‎, ‎w_2,\ldots,w_k\}$ of vertices and a‎ vertex $v$ in a connected graph $G$‎, ‎the ordered $k$-vector‎ ‎$r(v|W)=(d(v,w_1),d(v,w_2),\ldots,d(v,w_k))$ is called the‎ ‎(metric) representation of $v$ with respect to $W$‎, ‎where $d(x,y)$‎ ‎is the distance between the vertices $x$ and $y$‎. ‎The set $W$ is‎ ‎called a resolving set for $G$ if distinct vertices of $G$ have‎ ‎distinct representations with respect to $W$‎. ‎The minimum‎ ‎cardinality of a resolving set for $G$ is its metric dimension‎, ‎and a resolving set of minimum cardinality is a basis of $G$‎. ‎Lower bounds for metric dimension are important‎. ‎In this paper‎, ‎we investigate lower bounds for metric dimension‎. ‎Motivated by a lower bound for the metric dimension $k$ of a graph‎ ‎of order $n$ with diameter $d$ in [S‎. ‎Khuller‎, ‎B‎. ‎Raghavachari‎, ‎and‎ ‎A‎. ‎Rosenfeld‎, ‎Landmarks in graphs‎, ‎Discrete Applied Mathematics‎ ‎$70(3) (1996) 217-229$]‎, ‎which states that $k \geq n-d^k$‎, ‎we characterize‎ all graphs‎ ‎with this lower bound and obtain a new lower bound‎. ‎This new bound is better than the previous one‎, ‎for graphs with diameter more than $3$‎.

Two vertices u, v in a connected graph G are doubly resolved by vertices x, y of G if d(v, x) − d(u, x) 6= d(v, y) − d(u, y). A set W of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of W . Doubly resolving number of a graph G, denoted by ψ(G), is the minimum cardinality of a doubly resolving set for the graph G. The aim of this paper is to investigate doubly resolving sets in graphs. An upper bound for ψ(G) is obtained in terms of order and diameter of G. ψ(G) is computed for some graphs and all graphs G of order n with the property ψ(G) = n− 1 are determined. Also, doubly resolving sets for unicyclic graphs are studied and it is proved that the difference between the number of leaves and doubly resolving number of a unicyclic graph is at most 2.

For a set [Formula: see text] of vertices and a vertex [Formula: see text] in a graph [Formula: see text], the [Formula: see text]-vector [Formula: see text] is the adjacency representation of [Formula: see text] with respect to [Formula: see text], where [Formula: see text] and [Formula: see text] is the minimum of [Formula: see text] and the distance between the vertices [Formula: see text] and [Formula: see text]. The set [Formula: see text] is an adjacency resolving set for [Formula: see text] if distinct vertices of [Formula: see text] have distinct adjacency representations with respect to [Formula: see text]. The minimum cardinality of an adjacency resolving set for [Formula: see text] is its adjacency dimension. It is clear that the adjacency dimension of an [Formula: see text]-vertex graph [Formula: see text] is between [Formula: see text] and [Formula: see text]. The graphs with adjacency dimension [Formula: see text] and [Formula: see text] are known. All graphs with adjacency dimension [Formula: see text], and all [Formula: see text]-vertex graphs with adjacency dimension [Formula: see text] are studied in this paper. In terms of the diameter and order of [Formula: see text], a sharp upper bound is found for adjacency dimension of [Formula: see text]. Also, a sharp lower bound for adjacency dimension of [Formula: see text] is obtained in terms of order of [Formula: see text]. Using these two bounds, all graphs with adjacency dimension 2, and all [Formula: see text]-vertex graphs with adjacency dimension [Formula: see text] are characterized.

For a set W of vertices and a vertex v in a graph G, the k-vector r2(v|W) = (aG(v,w1),...,aG(v,wk)) is the adjacency representation of v with respect to W, where W = {w1,...,wk} and aG(x,y) is the minimum of 2 and the distance between the vertices x and y. The set W is an adjacency resolving set for G if distinct vertices of G have distinct adjacency representations with respect to W. The minimum cardinality of an adjacency resolving set for G is its adjacency dimension. It is clear that the adjacency dimension of an n-vertex graph G is between 1 and n-1. The graphs with adjacency dimension 1 and n-1 are known. All graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n-2 are studied in this paper. In terms of the diameter and order of G, a sharp upper bound is found for adjacency dimension of G. Also, a sharp lower bound for adjacency dimension of G is obtained in terms of order of G. Using these two bounds, all graphs with adjacency dimension 2, and all n-vertex graphs with adjacency dimension n-2 are characterized.

A set W \subseteq V (G) is called a resolving set, if for each pair of distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality of a minimum resolving set for G is called the metric dimension of G and is denoted by dim_M(G). A k-tree is a chordal graph all of whose maximal cliques are the same size k + 1 and all of whose minimal clique separators are also all the same size k. A k-path is a k-tree with maximum degree 2k, where for each integer j, k \leq j < 2k, there exists a unique pair of vertices, u and v, such that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then dim_M(G) = k. Moreover, we provide a characterization of all 2-trees with metric dimension two.

A set W subset of V (G) is called a resolving set, if for each two distinct vertices u, v is an element of V(G) there exists w is an element of W such that d(u,w) not equal d(v, w), where d(x, y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a unique basis graph. In this paper, we study some properties of unique basis graphs.

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),.,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. A minimum resolving set for $G$ is a basis of $G$ and its cardinality is the metric dimension of $G$. The resolving number of a connected graph $G$ is the minimum $k$, such that every $k$-set of vertices of $G$ is a resolving set. A connected graph $G$ is called randomly $k$-dimensional if each $k$-set of vertices of $G$ is a basis. In this paper, along with some properties of randomly $k$-dimensional graphs, we prove that a connected graph $G$ with at least two vertices is randomly $k$-dimensional if and only if $G$ is complete graph $K_{k+1}$ or an odd cycle.

A set W subset of V(G) is called a resolving set for G, if for each two distinct vertices u, v is an element of V(G) there exists w is an element of W such that d(u, w) not equal d(v, w), where d(x, y) is the distance between the vertices x and y. The minimum cardinality of a resolving set for G is called the metric dimension of G, and denoted by dim(G). In this paper, it is proved that in a connected graph G of order n which has a cycle, dim(G) <= n - g(G) + 2, where g(G) is the length of the shortest cycle in G, and the equality holds if and only if G is a cycle, a complete graph or a complete bipartite graph K-s,K-t, s,t >= 2.

For an ordered set $W=\{w_1,w_2,...,w_k\}$ of vertices and a vertex $v$ in a connected graph $G$, the ordered $k$-vector $r(v|W):=(d(v,w_1),d(v,w_2),...,d(v,w_k))$ is called the (metric) representation of $v$ with respect to $W$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The set $W$ is called a resolving set for $G$ if distinct vertices of $G$ have distinct representations with respect to $W$. The minimum cardinality of a resolving set for $G$ is its metric dimension. In this paper, we characterize all graphs of order $n$ with metric dimension $n-3$.

For an ordered set W = w1,w2,...,wk of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W) := (d(v,w1),d(v,w2),...,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we investigate the metric dimension of the lexicographic product of graphs G and H, G[H] for some known graphs.

A set W⊆ V(G) is called a resolving set, if for each two distinct vertices u,v∈ V(G) there exists w∈ W such that d(u,w)≠ d(v,w), where d(x,y) is the distance between the vertices x and y. A resolving set for G with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we study some properties of uniquely dimensional graphs.

A set $W\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $\beta(G)$. In this paper, it is proved that in a connected graph $G$ of order $n$ which has a cycle, $\beta(G)\leq n-g(G)+2$, where $g(G)$ is the length of a shortest cycle in $G$, and the equality holds if and only if $G$ is a cycle, a complete graph or a complete bipartite graph $K_{s,t}$, $ s,t\geq 2$.

For an ordered set W = {w1, w2, . . . , wk} of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W ) := (d(v, w1), d(v, w2), . . . , d(v, wk)) is called the (metric) representation of v with respect to W , where d(x, y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W . The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we study the metric dimension of the composition product of graphs G and H, G[H]. First, we introduce a new parameter which is called adjacency metric dimension of a graph. Then, we obtain the metric dimension of G[H] in terms of the order of G and the adjacency metric dimension of H.

A set W⊆ V(G) is called a resolving set, if for each two distinct vertices u,v∈ V(G) there exists w∈ W such that d(u,w)≠ d(v,w), where d(x,y) is the distance between the vertices x and y. The minimum cardinality of a resolving set for G is called the metric dimension of G, and denoted by β(G). In this paper, we prove that in a connected graph G of order n, β(G)≤ n-γ(G), where γ(G) is the domination number of G, and the equality holds if and only if G is a complete graph or a complete bipartite graph K_s,t, s,t≥ 2. Then, we obtain new bounds for β(G) in terms of minimum and maximum degree of G.