Generalized Cospectral Mates

• Generalized cospectral mates are non-isomorphic graphs with identical adjacency and complement spectra, exemplified by constructions in the Johnson scheme.
• They are generated using algebraic methods like Godsil–McKay switching, which leverages regular partitions and combinatorial symmetries to produce cospectral variants.
• These graphs highlight limitations of spectral invariants in distinguishing non-isomorphic structures and open avenues for research in algebraic combinatorics and graph isomorphism.

A generalized cospectral mate of a graph is a non-isomorphic graph whose adjacency matrix shares both the spectrum of the original graph and the spectrum of its complement. This notion is centrally linked to the theory of spectral graph invariants, association schemes, and the study of graph isomorphism through spectral properties. Generalized cospectral mates in the context of highly regular graph classes, such as those arising in the Johnson association scheme, admit systematic constructions via algebraic methods such as Godsil-McKay switching, and their existence has implications on the uniqueness of spectral characterization within these combinatorial families. The interplay between combinatorial structure, spectral data, and algebraic symmetries underpin the framework and enumeration of generalized cospectral mates.

1. Definitions and the Johnson Association Scheme

Generalized cospectral mates are defined for graphs $G$ and $H$ on the same vertex set, typically through the adjacency matrices $A(G)$ and $A(H)$. For strict non-isomorphic generalized cospectral mates, $H$ must satisfy:

• $$\operatorname{Spec}(A(G)) = \operatorname{Spec}(A(H))$$,
• $$\operatorname{Spec}(A(\overline{G})) = \operatorname{Spec}(A(\overline{H}))$$,

where $\overline{G}$ denotes the complement.

In the Johnson scheme, the relevant graphs $J_S(n, k)$ are constructed as follows (Cioabă et al., 2017):

• Vertices: all $k$-subsets $A \subseteq [n] = \{1,2,\dots,n\}$,
• Edges: $\{A, B\}$ with $|A \cap B| \in S$ for $S \subseteq \{0, 1, \ldots, k-1\}$.

Special cases include the Johnson graph $J(n,k) = J_{\{k-1\}}(n,k)$ and the Kneser graph $K(n,k) = J_{\{0\}}(n,k)$.

The spectral decomposition of $J_S(n,k)$ is governed by the Johnson association scheme and the dual Hahn polynomials, ensuring highly structured common eigenspaces for all graphs in the scheme. This structure is crucial to both the possibility of cospectral mates and the application of algebraic switching methods (Cioabă et al., 2017).

2. Spectrum and Eigenvalue Multiplicities in $J_S(n, k)$

The spectrum of $J_S(n, k)$ is explicitly determined by the underlying association scheme. The adjacency matrix $A_S$ of $J_S(n, k)$ is

$A_S = \sum_{j \in S} A_j$

where $A_j$ are the basis adjacency matrices for the scheme, each corresponding to $|A \cap B| = k - j$.

On the $i$th eigenspace ($0 \leq i \leq k$):

• Multiplicities: $m_i = \binom{n}{i} - \binom{n}{i-1}$ (with convention $\binom{n}{-1}=0$).
• Eigenvalues: $\theta_i = \sum_{j=0}^m P_j(i)$, with $P_j(i)$ the dual Hahn polynomials:

$$P_j(i) = \sum_{t=0}^j (-1)^t \binom{i}{t} \binom{k-i}{j-t} \binom{n-k-i}{j-t}$$

For $S = \{0,1,\ldots,m\}$, the spectrum reduces to linear combinations of these polynomials. The extreme eigenvalues $\theta_0$ and $\theta_k$ admit combinatorial expressions.

The highly structured nature of these eigenspaces, uniquely determined by $n, k$, and $S$, both restricts and enables the construction of generalized cospectral mates via global combinatorial operations (Cioabă et al., 2017).

3. Godsil–McKay Switching and Construction of Cospectral Mates

Godsil–McKay switching is the principal methodology for constructing cospectral (and generalized cospectral) mates in regular graph families, especially Johnson-type schemes. The version relevant to the Johnson setting (Cioabă et al., 2017) is as follows:

Let $G = (V, E)$ be regular, $V = C \cup D$ a partition such that:

• $C$ induces a regular subgraph,
• Every $v \in D$ has $0$, $|C|/2$, or $|C|$ neighbors in $C$.

Define $G'$ by, for $v \in D$ with exactly $|C|/2$ neighbors in $C$, deleting those edges and joining $v$ to the other $|C|/2$ vertices of $C$. Then $G'$ is cospectral with $G$. This construction has algebraic underpinnings in automorphism group actions and equitable partitions.

For $J_S(n, k)$ with $S = \{0, 1, \ldots, m\}$:

• For $k = 2m+1$, $n \geq 4m+2$: Taking $C = \{c \subseteq [2m+2] : |c| = 2m+1\}$ forms a valid switching set (Cioabă et al., 2017).
• For $n = 3k - 2m - 1$, $k \geq m+2$: $$C = \{c \subseteq [3k-2m-1] : |c| = k, [k-1] \subseteq c\}$$ is admissible.

This yields explicit infinite families of non-isomorphic cospectral mates, and crucially, the mates constructed via such switching operations are demonstrably non-isomorphic by carefully tracking the invariance violations under common-neighbor statistics or automorphism-induced constraints (Cioabă et al., 2017, Abiad et al., 2023).

4. Main Results on Generalized Cospectral Mates in the Johnson Scheme

The key theorems for the existence and enumeration of generalized cospectral mates in $J_S(n,k)$ are as follows (Cioabă et al., 2017):

• Existence: For $m \geq 0$, $k \geq \max(m + 2, 3)$, the graphs $J_S(3k - 2m - 1, k)$ with $S = \{0, \ldots, m\}$ admit non-trivial cospectral (hence generalized cospectral) mates via Godsil–McKay switching.
• Broader family: For $m \geq 2$, $n \geq 4m + 2$, one can take $S = \{0, \ldots, m\}$ and $k = 2m + 1$ to construct cospectral mates.
• Non-isomorphism: The cospectral mates constructed by switching differ in specific combinatorial invariants (requiring isomorphisms to preserve these, which is violated), establishing their status as strict mates, not trivial relabelings.

A summary table for the main parametric infinite families is:

Parameter Regime	Universe Size $n$	$k$	Switching Set $C$	Cospectral Mate Exists
$k=2m+1$, $n \geq 4m+2$	$n \geq 4m+2$	$2m+1$	$C = \{c \subseteq [2m+2] : |c| = 2m+1\}$	Yes
$n = 3k-2m-1$, $k \geq m+2$	$3k-2m-1$	$k$	$C = \{c : |c|=k, [k-1] \subseteq c\}$	Yes

Additionally, computational searches for $k \leq 5$ and small $n$ confirm the sharpness of these constructions and, in some cases, generate new small switching sets (Cioabă et al., 2017).

5. Computational and Experimental Results

Systematic computer searches using exhaustive and backtracking techniques have extended these switching constructions to all graphs $J_S(n, k)$ with $k \leq 5$. Findings (Cioabă et al., 2017) include:

• For $J_{\{0,1\}}(9,4)$, new switching sets of size 4 recover the infinite families predicted by the main theorems.
• For $K(9,3)$, $K(10,3)$, $K(11,3)$, $K(12,3)$, and $K(10,4)$, no new switching sets of size 4 or 6 were found beyond what was known from prior work.
• For $J_{\{2\}}(8,4)$, switching sets of size 4 yield isomorphic mates, but switching sets of size 8 (e.g., comprised of two 4-cycles or 6-regular 8-vertex graphs) yield non-isomorphic cospectral mates.

These results demonstrate that the switching method is both comprehensive and, in some regimes, exhaustive for generating generalized cospectral mates in the Johnson association scheme for small parameters (Cioabă et al., 2017). The associated codebase is publicly available.

6. Connections to Broader Spectral Uniqueness and Open Problems

The existence of generalized cospectral mates constructed via Godsil–McKay switching demonstrates that many naturally occurring, highly regular graphs are not determined by spectrum or generalized spectrum. This sharply contrasts with the situation for many random graphs or certain highly asymmetric families where the spectrum is conjectured (or proven) to be a complete invariant barring isomorphism.

Key open problems and research directions include:

• Classification of all $J_S(n, k)$ (or union-of-classes graphs) determined by spectrum versus those admitting generalized cospectral mates, especially for sporadic cases and for larger $k$.
• Extension of switching techniques to the $q$-analog Grassmann schemes and Kneser graphs, where early evidence suggests similar infinite families can be constructed via $q$-analogs of the switching paradigms (Abiad et al., 2023).
• Systematic tabulation and theoretical understanding of the switching sets for arbitrary unions of classes in the Johnson scheme, possibly with computational assistance for larger $n$ and $k$.

The interplay between combinatorial design, algebraic automorphisms, and spectral invariant theory is central in furthering the classification of generalized cospectral mates in association schemes and related algebraic graph frameworks.

7. Significance of Generalized Cospectral Mates in Algebraic Graph Theory

The Johnson association scheme and related structures (Kneser, Hamming, Grassmann schemes) serve as crucial testing grounds for ideas in algebraic and spectral graph theory. The construction and classification of generalized cospectral mates illuminate not only the limitations of spectral invariants for distinguishing non-isomorphic graphs but also the deep connections between combinatorial partitions, switching operations, association schemes, and spectral algebra (Cioabă et al., 2017, Abiad et al., 2023).

The analytical tools developed—polynomial eigenvalue expressions, explicit constructions of switching sets, combinatorial and computational validation of non-isomorphism—constitute essential techniques in modern algebraic combinatorics, with implications for the design and analysis of networks, quantum walks, and graph isomorphism testing in highly regular families.