Adjacency-Diametrical Matrix (AD Matrix)

• The adjacency-diametrical matrix is a hybrid graph descriptor that combines immediate adjacency with farthest (diametrical) distance measures to capture both local and global structure.
• It represents connections with weighted entries—using 1 for adjacent vertices and the graph diameter for diametrically distant pairs—facilitating comprehensive network analysis.
• Its spectral properties and determinant formulations yield valuable insights into graph invariants, supporting advanced applications in network comparison and isomorphism testing.

The adjacency-diametrical matrix (AD matrix) is a distance-weighted vertex-pair descriptor for simple, connected graphs, unifying adjacency and extremal distance information into a single $n \times n$ matrix. For a graph $G$ with vertex set $V=\{v_1,\dots,v_n\}$ and diameter $d$, the $(i,j)$ entry of $\mathrm{AD}(G)$ is $1$ if $v_i$ and $v_j$ are adjacent, $d$ if they are at diametrical distance, and $0$ otherwise. This construction synthesizes adjacency matrices, distance matrices, and higher-order combinatorial topology, subsuming key statistics and network invariants via explicit algebraic and spectral analysis (Amruthavarshini et al., 3 Jan 2026). Variants such as the neighbor (AD) matrix encode histogram distance distributions (Roginski et al., 2015), and the spectral excess paradigm connects AD matrices to polynomial expressions in the adjacency algebra (Fiol et al., 2019).

1. Mathematical Definition and Structure

Let $G=(V,E)$ be a connected graph of order $n$ and diameter $d$. The adjacency-diametrical matrix $\mathrm{AD}(G)$ is defined by

$$(\mathrm{AD}(G))_{ij} = \begin{cases} 1, & \text{if } d_G(v_i,v_j)=1, \ d, & \text{if } d_G(v_i,v_j)=d, \ 0, & \text{otherwise}, \end{cases}$$

where $d_G(v_i, v_j)$ denotes the graph distance. That is,

$\mathrm{AD}(G) = A_1(G) + d \cdot A_d(G),$

where $A_k(G)$ is the 0-1 matrix marking vertex pairs at distance $k$. For $d=2$, $\mathrm{AD}(G)$ coincides with the customary (unweighted) distance matrix, while for $d=1$ it reduces to the adjacency matrix (Amruthavarshini et al., 3 Jan 2026).

The neighbor (AD) matrix $M\in\mathbb{N}^{n\times d}$ is defined by

$$M_{i,k} = \left| \{ v_j\in V : d_G(v_i,v_j)=k \} \right|, \qquad k=1,\ldots,d, \quad i=1,\ldots,n,$$

encoding for each vertex the number of neighbors at every possible distance (Roginski et al., 2015).

2. Spectral Properties of AD Matrices

Explicit spectra have been determined for several graph families (Amruthavarshini et al., 3 Jan 2026):

• Paths $P_n$: The characteristic polynomial of $\mathrm{AD}(P_n)$ is

$$\varPhi_{\mathrm{AD}(P_n)}(x) = \varPhi_n(x) - (n-1)^2\,\varPhi_{n-2}(x) + 2(1-n),$$

where $\varPhi_k(x)$ is the characteristic polynomial of the path adjacency matrix.

• Cycles $C_n$: If $n$ is even (diameter $d=n/2$),

$$\lambda_k = 2\cos(2\pi k/n) + (n/2)(-1)^k, \quad k=0,\ldots,n-1,$$

and if $n$ is odd,

$$\lambda_k = 2\cos(2\pi k/n) + (n-1)(-1)^k\cos(\pi k/n).$$

• Double-star graphs $S_{p,q}$: The characteristic polynomial is

$$x^{n-4}\left[x^4 - (9n_1 n_2 - 8n_1 - 8n_2 + 8)x^2 + 4(n_1 n_2 - n_1 - n_2 + 1)\right]$$

with $n_1=p+1$, $n_2=q+1$, $n=n_1+n_2$, diameter $d=3$.

For graph products, the AD spectrum transforms according to explicit rules. For Cartesian products of distance-regular graphs $G$ and $H$, one has

$$\mathrm{Spec}(\mathrm{AD}(G \square H)) = \{ \lambda_i + \mu_j + (d_G + d_H)\gamma_i\delta_j : 1\leq i\leq |G|, 1\leq j \leq |H| \},$$

where the $\lambda_i$, $\mu_j$ are adjacency eigenvalues, and $\gamma_i$, $\delta_j$ are co-eigenvalues for the respective diametrical adjacency operators.

3. Determinant Formulation and Combinatorial Interpretation

The determinant of $\mathrm{AD}(G)$ admits a partition-based expansion (Amruthavarshini et al., 3 Jan 2026). For an adjacency-diametrical partition $\mathcal{S}$ organizing $V(G)$ into pairwise adjacent/antipodal pairs and adjacency-diametrical cycles, one has

$$\det\,\mathrm{AD}(G) = \sum_{\mathcal{S}} (-1)^{n-p(\mathcal{S})-p_1(\mathcal{S})} 2^{p_1(\mathcal{S})} d^{2a(\mathcal{S}) + a_1(\mathcal{S})},$$

with $p(\mathcal{S})=$ number of two-vertex parts, $p_1(\mathcal{S})=$ number of cycles of length at least $3$, $a(\mathcal{S})=$ number of antipodal pairs, and $a_1(\mathcal{S})=$ number of antipodal edges in cycles.

For example, $P_4$ has determinant $4$, exactly matching the combinatorial partition analysis. This expansion encapsulates contributions from both adjacency and diametrical relationships.

4. Bipartiteness, Characterization, and Polynomials

A graph $G$ is diametrical bipartite if it is bipartite and no two vertices in the same part are at diametrical distance. This occurs precisely when the diameter $d$ is odd (Amruthavarshini et al., 3 Jan 2026). The following conditions are equivalent for connected $G$:

(i) $G$ is diametrical bipartite. (ii) The weighted graph $\mathscr{G}_G$ defined by $\mathrm{AD}(G)$ is bipartite. (iii) Every adjacency-diametrical cycle has even length.

The spectrum of $\mathrm{AD}(G)$ in this case is symmetric about zero; all odd-index coefficients in the characteristic polynomial vanish. This criterion provides a spectral and combinatorial tool for bipartite characterization.

5. Bounds and Extremal Eigenvalues

Let $$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$$ denote the eigenvalues of $\mathrm{AD}(G)$, $m=|E|$ the number of edges, and $\hat{d}(v)$ the diametrical degree of vertex $v$ (number of vertices at distance $d$).

Key spectral facts (Amruthavarshini et al., 3 Jan 2026):

• $$\sum_{i=1}^n \lambda_i = \mathrm{tr}(\mathrm{AD}(G)) = 0$$
• $\sum_{i=1}^n \lambda_i^2 = 2m + d^2 \hat{d}(G)$

The spectral radius obeys

$$\lambda_1 \leq \sqrt{\frac{n-1}{n} (2m + d^2 \hat{d}(G))},$$

and for diametrical bipartite $G$, the smallest eigenvalue satisfies

$$|\lambda_n| \leq \sqrt{m + \frac{d^2}{2} \hat{d}(G)}.$$

Bounds in terms of the minimum/maximum diametrical degree $\hat{\delta}(v) = \deg(v) + d \cdot \hat{d}(v)$ are tight for AD-regular graphs.

6. Relationship to Neighbor Matrix, Distance Matrices, and Algebraic Framework

The neighbor matrix $M$ defined by $M_{i,k}$ (the number of vertices at distance $k$ from $v_i$) encodes all frequency data for distance distributions up to the diameter (Roginski et al., 2015). It generalizes the adjacency matrix (column $k=1$), the degree sequence, and contains key graph invariants:

• Diameter, radius, center, periphery
• Number of edges, graph density
• Closeness centrality: $CC(v_i) = \frac{n-1}{\sum_{k=1}^d k M_{i,k}}$
• Average distance (Wiener index divided by $\binom{n}{2}$): $$\frac{1}{n(n-1)} \sum_{i=1}^n \sum_{k=1}^d k M_{i,k}$$

$M$ provides a histogram of local-global distance structure, allowing both fine-grained and summary comparison of graph topology.

The spectral excess theorem (Fiol et al., 2019) connects AD matrices and the adjacency algebra of regular graphs: the diametrical matrix $A_D$ is a polynomial in $A$ precisely when arithmetic/harmonic means of vertex excesses attain equality with certain predistance polynomial sums. In distance-regular graphs, this polynomial representation is unique, and the spectrum of $A_D$ derives from the predistance polynomials evaluated at adjacency eigenvalues.

7. Applications and Structural Insights

AD and neighbor matrices support advanced graph comparison, serving as discriminators when classical invariants fail. For graph isomorphism, a mismatch in the AD matrix immediately rules out equivalence; matched matrices signal identical distance distributions (Roginski et al., 2015). Norm-based matrix comparisons (e.g., Frobenius norm) quantify topological “distance” between graphs.

Vertex centrality and structural influence are refined by recomputing $M$ post-removal, measuring deviation in the distance matrix. Large changes correspond to vertices pivotal for short paths, sharpening the identification of topologically significant vertices.

Under graph operations, the AD matrix transforms in structured fashion, yielding explicit spectra for joins, lexicographic products (via tensor-product decompositions), and Cartesian products given distance-regularity (Amruthavarshini et al., 3 Jan 2026). This behavior enables analysis of composite networks and facilitates the study of spectral and combinatorial invariants in expanded graph classes.

In summary, the adjacency-diametrical matrix is a foundational descriptor synthesizing adjacency, extremal, and distributional graph properties, serving both theoretical characterization and practical network analysis across families of graphs and graph operations.