Metric basis and dimension of barycentric subdivision of zero divisor graphs

Let $R$ be a commutative ring with unity 1, and $ G(V,E)$ be a simple, connected, nontrivial graph. Let $d(a,c)$ be the distance between the vertices $a$ and $c $ in $G$. An undirected zero divisor graph of a ring $R$ is denoted by $Γ(R) = (V(Γ(R)), E(Γ(R)))$, where the vertex set $V(Γ(R))$ consists of all the non-zero zero-divisors of $R$, and the edge set $E(Γ(R))$ is defined as follows: $E(Γ(R)) = $ ${e = a_1a_2$ $ |$ $ a_1 \cdot a_2 = 0$ $&$ $ a_1, a_2 \in V(Γ(R))}$. In this article, we consider the zero divisor graph of a group of integers modulo (n), denoted as (Γ(\mathbb{Z}_n)), where (n=pq). Here, (p) and (q) are distinct primes, with (q > p). We aim to determine the metric dimension of the barycentric subdivision of the zero divisor graph (Γ(\mathbb{Z}_n)), denoted by (dim(BS(Γ(\mathbb{Z}_n)))), and we also prove that (dim(BS(Γ(\mathbb{Z}_n)))\geq q-2) for every (n=pq), where (p) and (q) are distinct primes and $q>p$.

💡 Research Summary

The paper investigates the metric dimension of the barycentric subdivision of zero‑divisor graphs arising from the ring ℤₙ where n is the product of two distinct primes, n = p q with q > p. A zero‑divisor graph Γ(R) has as vertices the non‑zero zero‑divisors of a commutative ring R, and an edge joins two vertices a and b precisely when a·b = 0 in R. The metric dimension dim(G) of a graph G is the size of the smallest resolving set—a set of reference vertices such that every vertex of G is uniquely identified by its vector of distances to the reference vertices.

The authors first recall basic graph‑theoretic notions (neighbourhood, independent set, resolving set, metric basis) and known results on metric dimension for paths, trees, and complete bipartite graphs. They then focus on the specific structure of Γ(ℤₙ) when n = p q. In this case the non‑zero zero‑divisors split into two natural parts: multiples of p (p, 2p, …, (p (q − 1))) and multiples of q (q, 2q, …, (q (p − 1))). Edges exist only between a multiple of p and a multiple of q, so Γ(ℤₙ) is essentially a complete bipartite graph K_{q‑1, p‑1} possibly with isolated vertices removed.

The barycentric subdivision BS(Γ(ℤₙ)) is obtained by inserting a new vertex into every edge of Γ(ℤₙ). Consequently each original edge becomes a path of length two, and the resulting graph is bipartite and loop‑free. The authors analyse BS(Γ(ℤₙ)) for three families of (p, q):

1. p = 2: BS(Γ(ℤ_{2q})) is a tree with 2q − 1 vertices and 2q − 2 edges. Using the known formula for the metric dimension of a tree (dim(T) = ∑_{v}(l_v − 1) where l_v counts the number of “legs” attached to v), they obtain dim = q − 2.

2. p = 3: The subdivision yields a more intricate four‑partite structure (sets A, B₁, C₁, Q). The authors explicitly construct a candidate resolving set
   E = {a₁, b₁₂, b₁₃, …, b₁_{q‑2}}
   and compute distance vectors δ(v | E) for every vertex v. They show that all vectors are distinct, establishing dim ≤ q − 2. To prove the lower bound, they argue that the set A contains q − 1 mutually non‑adjacent vertices; any resolving set with fewer than q − 2 vertices would leave at least two vertices of A indistinguishable, contradicting the definition of a resolving set. Hence dim = q − 2.

3. General odd primes (p ≥ 5, q ≥ 2p − 1): The subdivision is partitioned into six families (A, B, C, and several sub‑families indexed by ν). A more elaborate resolving set E is defined, mixing vertices from A, selected vertices from the C‑family, and a block of vertices from the B‑family. Again, exhaustive distance‑vector calculations demonstrate that every vertex outside E has a unique representation, giving dim ≤ q − 2. The lower bound follows from the same independence argument applied to the large independent set A: any smaller resolving set would fail to separate at least two vertices of A, forcing dim ≥ q − 2. Consequently dim = q − 2 for this whole class.

The main theorems are:

• Theorem 3.1: For p = 2 or p = 3, dim(BS(Γ(ℤ_{pq}))) = q − 2.
• Theorem 3.2: For odd primes p ≥ 5 and q ≥ 2p − 1, dim(BS(Γ(ℤ_{pq}))) = q − 2.
• Corollary: For any n = p q with distinct primes p, q (q > p), dim(BS(Γ(ℤₙ))) ≥ q − 2.

The proofs rely on two complementary strategies: (i) constructing explicit small resolving sets and verifying uniqueness of distance vectors, and (ii) leveraging the size of the independent set A to establish a necessary lower bound. The barycentric subdivision is crucial because it transforms each original edge into a two‑step path, thereby providing additional “mid‑edge” vertices that increase the granularity of distance measurements without dramatically inflating the graph’s size.

The paper’s contributions are significant for several reasons. First, it provides an exact formula for the metric dimension of a whole family of subdivided zero‑divisor graphs, a problem that had previously been addressed only for special cases (e.g., paths, cycles, complete bipartite graphs). Second, the result shows that the metric dimension depends solely on the larger prime q, regardless of the smaller prime p (provided q is sufficiently large relative to p). This insight simplifies the analysis of large rings where p and q may be far apart. Third, the methodology—explicit distance‑vector calculations combined with independence arguments—offers a template for tackling metric dimension problems in other graph transformations (e.g., line graphs, power graphs).

However, the paper leaves open the case where q < 2p − 1. The authors’ technique for the general odd‑prime case hinges on the inequality q ≥ 2p − 1 to guarantee enough vertices in certain sub‑families, and they do not address whether dim could exceed q − 2 when this condition fails. Moreover, the paper does not explore algorithmic aspects (e.g., complexity of finding a metric basis in these graphs) or potential applications such as network navigation or fault‑tolerant routing, which are natural extensions given the relevance of metric dimension in those domains.

In summary, the article delivers a clear, rigorous determination of the metric dimension for barycentric subdivisions of zero‑divisor graphs of ℤ_{pq}, establishing dim = q − 2 for a broad range of prime pairs and providing both constructive upper bounds and combinatorial lower bounds. The work bridges algebraic graph theory and metric graph invariants, and it opens avenues for further investigation of metric dimensions under other graph operations and for rings with more complex factorisations.