Minimum-Distance Graphs

• Minimum-distance graphs are defined as the graphs connecting codewords or elements that differ by the minimal metric gap, revealing key structural and symmetry properties.
• They enable reconstructibility and isomorphism testing by mapping maximum cliques to coordinate features and ensuring code equivalence.
• These graphs also inform optimization problems, such as designing pairwise distance preservers and analyzing spectral characteristics in network structures.

A minimum-distance graph encodes, for a code or combinatorial structure, the relationships between codewords or elements that are separated by the minimal possible distance according to a prescribed metric. In coding theory, these graphs are fundamental for characterizing the fine algebraic and combinatorial structure of codes, reconstructing codes from partial information, and analyzing isomorphism invariants. In other contexts such as distance preservers in combinatorial optimization or spectral graph invariants, the notion of minimum or pairwise distance also plays a significant analytic and algorithmic role.

1. Minimum-Distance Graphs in Coding Theory

Let $C \subseteq \mathbb{F}_2^n$ be a binary code with minimum Hamming distance $d$. The minimum-distance graph $G(C)$ is the simple graph whose vertex set is $C$, with two vertices adjacent if and only if the associated codewords differ in exactly $d$ positions. Formally,

$$G(C) = (V,E),\quad V = C,\quad E = \{\{x,y\} \subset C \mid d_H(x,y) = d\}$$

Edges trace out the codeword pairs at the tightest possible separation, providing a graph-theoretic visualization of the code's "nearest neighbors." For extended 1-perfect codes ($d=4$), the structure of $G(C)$ reflects the induced Steiner quadruple system (SQS) design, with maximal cliques corresponding to "points" (coordinates) of the design. The order of $G(C)$ is the code size $|C|$, the regular degree quantifies nearest neighbor count, and the clique and connectivity structure convey deep information about error-correcting capability and uniqueness properties (0810.5633).

2. Reconstructibility and Isomorphism Invariance

A principal application of minimum-distance graphs arises in code isomorphism and reconstructibility questions. For binary extended 1-perfect codes ($d=4$, $n \geq 16$), one can reconstruct the code—up to equivalence (coordinate permutation and translation)—from its minimum-distance graph. The reconstruction proceeds by identifying maximal cliques as coordinate stars of the associated SQS, then inductively recovering codewords by their pattern of intersections. This result, due independently to Mogilnykh et al., underpins the theorem that two extended 1-perfect codes have isomorphic minimum-distance graphs if and only if they are equivalent as codes. Furthermore, automorphism groups of the code and of $G(C)$ are isomorphic—formally, $$\operatorname{Aut}(G(C)) \cong \operatorname{Aut}(C)$$, where the group action reflects both coordinate and combinatorial symmetries (0810.5633).

Analogous statements hold for ordinary (unextended) 1-perfect codes by considering the extended code and then puncturing the parity-check coordinate. For smaller code lengths ($n \leq 8$), uniqueness makes reconstructibility immediate, while for $n=15$ with many inequivalent codes, minimum-distance graphs remain a complete invariant.

3. Minimum-Distance Graphs for Preparata and Other Nonlinear Codes

For nonlinear codes such as the extended Preparata code $P(m)$ ($n=2^m$, $d=6$, $m$ even), the minimum-distance graph $G(P(m))$ connects codewords at Hamming distance 6. Each vertex has regular degree equal to the number $A_6$ of weight-6 codewords: $A_6 = \frac{\binom{n}{3} (n-4)}{20}$ Maximal cliques in the neighborhood of $0$ are those indexed by triples of coordinates, i.e., each triple $t \subset \{1,\ldots,n\}$, $|t|=3$, defines the clique

$$C(t) = \{v \in N(0) : t \subset \operatorname{supp}(v)\}$$

These cliques have size $n-4$ and any clique exceeding size 13 must be of this form for $m\geq 6$. The family of maximum cliques and their intersections encode the coordinate system of the code up to permutation. The minimum-distance graph is again a complete invariant of code equivalence: two extended Preparata codes have isomorphic minimum-distance graphs if and only if they are equivalent (0902.1351).

4. Minimum-Weight Pairwise Distance Preservers

Beyond algebraic coding, minimum-distance graphs motivate combinatorial optimization problems, notably in the design of pairwise distance preservers for weighted graphs. Given a weighted graph $G = (V,E,c)$ and a set of demand pairs $P$, a subgraph $G'$ is a pairwise distance preserver for $P$ if it preserves the shortest-path distance for every pair in $P$. The Minimum-Weight Pairwise Distance Preserver (MWPDP) problem seeks a preserver that minimizes total edge cost, constrained by exact distance preservation. The Cost-Sharing Pairwise Distance Preserver (CSPDP) asks for the subgraph maximizing net savings versus the sum of dist$(s,t)$ over all pairs, rewarding overlap among shortest paths (Abdolmaleki et al., 2020).

An $O(m^{1/2+\epsilon})$-approximation algorithm for CSPDP exists for weighted graphs, with nearly tight lower bounds under standard complexity assumptions. The method partitions edges by thickness (frequency across shortest paths), applies combinatorial and LP-based sparsification, and recursively reduces problem size. The reduction from LABEL-COVER$_{\max}$ establishes strong hardness for approximation. The theoretical framework unifies and sharpens understanding of edge-sharing protocols for distance preservation, framed in terms of minimum-distance or distance graphs.

5. Distance Laplacian Energy and Extremal Graphs

For a simple connected graph $G$ of order $n$, the distance matrix $D(G)$ records shortest-path distances. The distance Laplacian matrix is $$D^L(G) = \operatorname{diag}(\operatorname{Tr}_G(v_1),\ldots,\operatorname{Tr}_G(v_n)) - D(G)$$, where $\operatorname{Tr}_G(v)$ is the sum of distances from $v$ to all other vertices. The spectral invariant, distance Laplacian energy,

$$DLE(G) = \sum_{i=1}^n |\rho^L_i - \tfrac{2 W(G)}{n}|$$

with $\rho^L_i$ the eigenvalues of $D^L(G)$ and $W(G)$ the Wiener index, quantifies deviation from "distance regularity." Extremal results show, for instance, that the complete bipartite graph has minimum DLE among all connected bipartite graphs of fixed order, while the join $K_k \triangledown (K_t \cup K_{n-k-t})$ achieves the DLE minimum among graphs with given vertex connectivity $k$ (Ganie et al., 2021).

6. Illustrative Examples and Applications

In extended 1-perfect codes ($n=16$), the minimum-distance graph encapsulates a Steiner quadruple system SQS(16), with 16 underlying "points" recovered by maximum cliques of size 35 among the 140 blocks. For the extended Preparata code $P(4)$ ($n=16$), the minimum-distance graph has 2048 vertices, each degree 336, and the structure of maximal cliques reflects the algebraic constraints of the code and its embedding in the unique SQS(16). These constructions serve not only for isomorphism testing and code recovery, but also delineate intricate relations between design theory, combinatorics, and spectral graph theory.

7. Broader Significance and Theoretical Impact

Minimum-distance graphs provide a rigorous mechanism to transfer coding-theoretic and combinatorial data to graph-theoretical and spectral domains. Their reconstructibility properties make them natural tools for classification, isomorphism testing, and understanding automorphism groups. In optimization, they ground questions of network design with rigid distance constraints in applications such as transportation and communication systems. The deep connections to spectral graph invariants and extremal combinatorics further illustrate the centrality of minimum-distance graphs within modern discrete mathematics and theoretical computer science.

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References:

• Mogilnykh et al., "Reconstructing Extended Perfect Binary One-Error-Correcting Codes from Their Minimum Distance Graphs" (0810.5633)
• Fernández‐Córdoba & Phelps, "On the minimum distance graph of an extended Preparata code" (0902.1351)
• Chlamtáč et al., "Minimum Weight Pairwise Distance Preservers" (Abdolmaleki et al., 2020)
• Shikare et al., "On distance Laplacian energy in terms of graph invariants" (Ganie et al., 2021)