Natural Graph Matrices

• Natural graph matrices are matrix representations derived from finite graphs using fixed algebraic operations on the adjacency matrix.
• They provide a framework for analyzing spectral properties and combinatorial correspondences, integrating standard matrices like adjacency, Laplacian, and Seidel with modern variants.
• Their algebraic structure and closure properties aid in distinguishing random graphs from highly symmetric ones by leveraging spectral invariants.

A natural graph matrix is any matrix representation of a finite graph obtained via a fixed, algebraically defined sequence of elementary operations—matrix addition, conventional (dot) product, and Hadamard (entrywise) product—applied to the adjacency matrix. This systematic construction class encompasses all standard graph-associated matrices including the adjacency, Laplacian, Seidel, distance, and mesh matrices, as well as more recently introduced objects such as clique and neighbourhood matrices. The theoretical foundation provided by the double algebra framework establishes closure, spectral properties, and combinatorial correspondences, ultimately yielding a comprehensive dictionary between graph-theoretic and algebraic features.

1. Formal Definition and the Double Algebra Framework

Let $G$ be a finite graph with $n$ vertices, $A_G\in M_n(\mathbb{F})$ its adjacency matrix over a field $\mathbb{F}$, and $M_n(\mathbb{F})$ the $n\times n$ matrix algebra. A natural graph matrix assignment is a map $M: \{\text{$n$-vertex graphs}\} \rightarrow M_n(\mathbb{F})$ such that there exists a fixed double polynomial $p(x)$ in the free double algebra $\mathbb{F}\langle\langle x \rangle\rangle$—built from $x$, the matrix identities $1^\cdot, 1^\circ$, and the operations $\cdot$ (ordinary product), $\circ$ (Hadamard product), and $\mathbb{F}$-linear combinations—for which $M_G = p(A_G)$ for all $G$ of size $n$.

The double algebra structure consists of two associative products $\cdot$ and $\circ$, each with its own identity, making $M_n(\mathbb{F})$ a canonical double algebra. Evaluation of a double polynomial at $A_G$ proceeds by interpreting $x$ as $A_G$ and applying the prescribed sequence of algebraic operations.

This definition directly encompasses adjacency, Laplacian, Seidel, signless Laplacian, and distance matrices, as each is realized as $M_G = p(A_G)$ for suitable $p$ in $\mathbb{F}\langle\langle x \rangle\rangle$ (Xiang, 31 Jan 2026).

2. Principal Families of Natural Graph Matrices

Standard choices within this class include:

Matrix	Double Polynomial $p(x)$	Description
Adjacency $A_G$	$x$	Standard vertex adjacency
Seidel/Complement	$1^\circ - 1^\cdot - x$	Seidel/complement adjacency; $J$ all-ones, $I$ identity
Laplacian $L_G$	$(x\cdot x) \circ 1^\cdot - x$	Degree minus adjacency
Signless Laplacian $Q_G$	$(x\cdot x)\circ 1^\cdot + x$	Degree plus adjacency
Distance matrix	Rational in $(x+1^\cdot)$, $\cdot$, $\circ$	Encodes shortest-path distances; see (Xiang, 31 Jan 2026)
Mesh Matrix $Mesh(G,T_0)$	Constructed via $\langle Z_i, Z_j\rangle$	Cycles via spanning trees (Cappell et al., 2023)
Clique Matrix $Z$	$ZZ^T$ (incidence-generalization)	Cluster-encoded adjacency (Barber, 2012)
Neighbourhood Matrix	$A(D-A)$	Degree, first/second neighbourhoods (Karunakaran et al., 2019)

The class is closed under natural (polynomial) combinations: linear sums, entrywise and dot products, and the formation of complements.

3. Spectral Determination and the van Dam–Haemers Conjecture

A core motivation is whether a "sensible" matrix exists whose spectrum determines a generic graph (as posed by van Dam and Haemers). The double algebra framework gives a sufficient condition: If the double algebra generated by $A_G$ is all of $M_n(\mathbb{F})$, then there exists a natural matrix whose spectrum determines $G$ up to isomorphism.

For standard random graphs $G(n,1/2)$ and almost all large $n$, this condition is satisfied: the double algebra generated by $A_G$ is generically full. Thus, there exists a single (explicit) natural graph matrix whose spectrum distinguishes random graphs up to isomorphism (Xiang, 31 Jan 2026).

However, for classes with small double algebra—distance-regular or strongly regular graphs—natural spectra cannot distinguish non-isomorphic graphs with the same intersection parameters. This limitation recapitulates why adjacency/Laplacian/Seidel spectra often fail for highly symmetric graphs.

4. Algebraic-Combinatorial Correspondences

The natural graph matrix framework is reflected and extended in the general study of graphical matrix spaces. For a bipartite or directed graph $G$:

• $S_G$ is the matrix space with zeros outside edges/arcs of $G$.
• Classical theorems (Tutte, Dieudonné, Flanders, Meshulam) match the maximum dimension of singular/nil/reducible subspaces in $S_G$ to combinatorial parameters (non-perfect-matchings, acyclic subgraphs, reducibility, etc.).
• Three fundamental correspondences are established: acyclicity $\leftrightarrow$ nilpotency, strong connectivity $\leftrightarrow$ irreducibility, isomorphism $\leftrightarrow$ conjugacy/congruence (Li et al., 2022).

This algebraic viewpoint recovers classical and modern results in both combinatorics and algebra, including Gerstenhaber's theorem on nilpotency, connections to quantum information theory, and open problems in symbolic computation.

5. Structural and Algorithmic Extensions

Several modern families generalize or interpolate between the classical natural graph matrices:

Mesh Matrix and Laplacian: $Mesh(G,T_0)$, defined via fundamental cycles determined by a spanning tree $T_0$, naturally generalizes the adjacency and Laplacian matrices. Its characteristic polynomial encodes tree/cycle enumeration via Tutte-type recursions. The associated mesh Laplacian $A(G,T_0)$ encompasses the Kirchhoff Laplacian as a special case and facilitates new spectral/geometric estimates (Cappell et al., 2023).

Clique Matrix: The $V\times C$ "clique matrix" $Z$ lists the membership of vertices in $C$ selected cliques, factorizing $A = H(ZZ^T)$. This representation naturally generalizes the adjacency and incidence matrices (the incidence matrix is the clique matrix for edge-cliques), enables latent-factor decompositions, and provides structured parameterizations for positive-definite matrices constrained by $G$'s sparsity pattern (Barber, 2012).

Neighbourhood Matrix: The matrix $\NM(G) = A(D-A)$ encodes degree, adjacency, and common-neighbourhood structure in a non-symmetric square matrix. It serves as a powerful tool for motif counting, two-level BFS, and efficient graph reconstruction (Karunakaran et al., 2019).

6. Spectral and Analytical Properties

Spectra of natural graph matrices are central: adjacency matrices yield symmetric spectra; Laplacians are positive semi-definite with spectral gap linked to connectivity. More generally, spectral relationships between $A$, $L$, and the normalized Laplacian $\mathcal{L}$ are quantitatively controlled by the degree spread $d_{max}-d_{min}$, sharply determining the consistency or divergence of inference under different spectral representations (Lutzeyer et al., 2017).

Mesh matrices ($Mesh(G,T_0)$), clique matrices, and neighbourhood matrices each introduce new spectral characteristics—Mesh matrices are always positive-definite with all eigenvalues at least $1$, while clique and neighbourhood matrices (often non-symmetric) require non-Hermitian spectral tools.

For random or generic graphs, the full matrix algebra property ensures that the spectrum of suitably constructed natural matrices is a strong graph invariant.

7. Applications and Connections to Broader Contexts

The natural graph matrix paradigm is foundational for:

• Spectral Clustering and Graph Signal Processing: The choice of matrix representation (adjacency, Laplacian, normalized Laplacian) directly impacts eigengaps, thus influencing clustering and filtering (Lutzeyer et al., 2017).
• Combinatorial Enumeration: Mesh and Laplacian matrices enumerate spanning trees/forests via matrix-tree and all-minors theorems (Cappell et al., 2023).
• Latent Structure Discovery: Clique matrices enable structured cluster detection and serve in sparse covariance estimation (Barber, 2012).
• Algorithmic Graph Analysis: Neighbourhood matrices facilitate efficient motif counting and subgraph exploration (Karunakaran et al., 2019).
• Quantum Information Theory: Algebraic-graph correspondences underlie the theory of quantum channels and zero-error quantum codes (Li et al., 2022).
• Computational Complexity: The connection of symbolic determinant/nilpotency testing to combinatorial matchings, acyclicity, and reducibility leads to open complexity-theoretic problems (Li et al., 2022).

In sum, natural graph matrices unify and expand the toolkit available for encoding, analyzing, and exploiting the combinatorial, spectral, and algorithmic properties of graphs, blending classical concepts with modern algebraic and statistical theory.