Cayley Sum Graphs: Structure & Applications

• Cayley sum graphs are defined on group elements where vertices x and y are adjacent if their sum lies in a prescribed subset, with variants in additive and multiplicative contexts.
• They admit detailed structural decompositions, including coset-level analysis and spectral expansion, leading to applications in pseudorandomness, expander families, and perfect codes.
• Their study bridges algebraic graph theory, additive combinatorics, and coding theory, uncovering new behaviors such as Ramanujan phenomena and precise coding-theoretic classifications.

A Cayley sum graph is a graph whose vertex set is a group $G$, with adjacency determined by a prescribed subset through a sum-type relation rather than the quotient or difference relation used in an ordinary Cayley graph. In additive notation, the basic rule is that $x$ and $y$ are adjacent when $x+y\in S$; in multiplicative notation, the literature also uses the equivalent-looking but non-identical rules $xy\in S$ and $y=x^{-1}s$ for some $s\in S$. This family sits at the intersection of algebraic graph theory, additive combinatorics, and coding theory, and it has been studied from several angles: structural decompositions, spectral expansion, random pseudorandomness, and perfect or total perfect codes (Biswas et al., 2021, Zhang, 2022).

1. Definitions and competing conventions

The term Cayley sum graph is not completely uniform across the literature. For a finite abelian group $G$ written additively and a subset $A\subset G$, one common definition sets

$E=\{(x,y)\in G\times G:\ x+y\in A\},$

sometimes allowing $x$0 and hence loops, especially in probabilistic work (Konyagin et al., 2017). In simple-graph treatments on finite abelian groups, one instead requires distinct vertices and usually assumes that the connection set is square-free, meaning that no element of the connection set is of the form $x$1, so that loops are excluded (Ma et al., 2020).

For general finite groups, two multiplicative conventions occur. One defines adjacency by $x$2, typically with $x$3 assumed to be a square-free normal subset so that the resulting graph is simple and undirected (Shaw et al., 5 Sep 2025). Another defines adjacency from a vertex $x$4 to the vertices $x$5 with $x$6; in this convention, undirectedness is obtained under hypotheses such as closure of $x$7 under conjugation, and the same framework extends naturally to twisted variants (Biswas et al., 2019, Biswas et al., 2020).

Convention	Adjacency rule	Typical hypotheses
Additive abelian	$x$8	Distinct vertices; often square-free $x$9
Multiplicative product-form	$y$0	$y$1 normal and square-free
Multiplicative inverse-form	$y$2	$y$3 symmetric / conjugacy-stable in undirected settings

These conventions coincide in some abelian settings after reparametrization, but in nonabelian settings they lead to genuinely different graph classes. A persistent point of terminology is that Cayley sum also appears in polytope theory, where for polytopes $y$4 one writes

$y$5

a construction unrelated to Cayley sum graphs except by name (Hibi et al., 2018).

A basic constraint already appears in the abelian simple-graph model: if $y$6 is a finite abelian group of odd order, then every element is a square, so the only simple Cayley sum graph is the empty graph (Ma et al., 2020).

2. Structural models and subgroup-based constructions

A particularly tractable subclass is given by subgroup sum graphs. If $y$7 is a finite abelian group and $y$8, the subgroup sum graph $y$9 has vertex set $x+y\in S$0, and distinct vertices $x+y\in S$1 are adjacent when $x+y\in S$2. The closely related extended subgroup sum graph $x+y\in S$3 uses the condition $x+y\in S$4. These graphs form a large subclass of Cayley sum graphs, with connection set $x+y\in S$5 or $x+y\in S$6 respectively (Cameron et al., 2021).

Their structure admits an explicit coset-level decomposition. Writing $x+y\in S$7 and examining the cosets $x+y\in S$8, one obtains three types: pairs of cosets $x+y\in S$9 with $xy\in S$0, which assemble into complete bipartite graphs $xy\in S$1; cosets with $xy\in S$2 but without involutions, which induce complete graphs $xy\in S$3; and cosets containing involutions, which again behave like complete graphs but, in the non-extended subgroup sum graph, with a matching removed. This decomposition yields a detailed description of clique number, independence number, spectrum, domination number, and connectedness (Cameron et al., 2021).

Two consequences are especially notable. First, all subgroup sum graphs and extended subgroup sum graphs are perfect graphs. Second, $xy\in S$4 and $xy\in S$5 are connected if and only if $xy\in S$6. In the subgroup sum graph case, if $xy\in S$7 denotes the number of elements $xy\in S$8 with $xy\in S$9, then

$y=x^{-1}s$0

and the chromatic number agrees with the clique number by perfectness (Cameron et al., 2021).

This structured viewpoint also unifies several previously studied examples. In particular, prime sum graphs arise when $y=x^{-1}s$1 for a prime $y=x^{-1}s$2, so that the subgroup is imposed by multiplication by $y=x^{-1}s$3 on the ambient abelian group (Cameron et al., 2021).

3. Spectrum, expansion, and Ramanujan phenomena

Spectral analysis of Cayley sum graphs parallels that of ordinary Cayley graphs but with an important sign ambiguity. For the inverse-form convention $y=x^{-1}s$4, if $y=x^{-1}s$5 is symmetric and closed under conjugation, then the adjacency eigenvalues are, up to factors of $y=x^{-1}s$6,

$y=x^{-1}s$7

where $y=x^{-1}s$8 ranges over the irreducible complex representations of $y=x^{-1}s$9. The same character-sum formula extends to twisted Cayley graphs and twisted Cayley sum graphs, again up to sign patterns controlled by the twisting involution (Biswas et al., 2021).

This spectral correspondence has several consequences. Expansion in one of the four related graph classes—ordinary Cayley, Cayley sum, twisted Cayley, twisted Cayley sum—transfers to the others under the hypotheses used in that framework, and the paper exhibits new expander and Ramanujan families. One highlighted example is that Paley sum graphs $s\in S$0 are Ramanujan, and analogous constructions are given on $s\in S$1, $s\in S$2, symmetric groups, and alternating groups (Biswas et al., 2021).

A complementary line of work develops Cheeger-type inequalities. If $s\in S$3 is a non-bipartite $s\in S$4-regular Cayley sum expander with vertex Cheeger constant $s\in S$5, then every nontrivial eigenvalue $s\in S$6 of the normalized adjacency operator lies in

$s\in S$7

so expansion bounds the spectrum away from both $s\in S$8 and $s\in S$9 (Biswas et al., 2019). For twisted Cayley sum graphs $G$0, connected and undirected, an analogous result yields

$G$1

with higher-order variants depending on the order of the twist (Biswas et al., 2020).

More recently, the lower and upper spectral gaps were related directly. For an undirected, connected, non-bipartite Cayley sum graph of degree $G$2, if $G$3 is the smallest eigenvalue and $G$4 the second largest eigenvalue of the normalized adjacency operator, then

$G$5

Thus, a positive top spectral gap forces a positive bottom spectral gap with explicit degree dependence (Saha, 2023).

4. Random Cayley sum graphs and pseudorandomness

In the random model over a finite abelian group $G$6, one chooses a subset $G$7 by including each element independently with probability $G$8, and then forms the graph with edge relation $G$9. In this formulation loops are allowed, and the central question is not simplicity but induced-subgraph edge density (Konyagin et al., 2017).

The principal theorem shows that random Cayley sum graphs are pseudorandom on induced subgraphs whose size is only slightly superlogarithmic. Writing $A\subset G$0, if $A\subset G$1, then with probability $A\subset G$2,

$A\subset G$3

simultaneously for all $A\subset G$4 satisfying

$A\subset G$5

The stated asymptotic implies that induced subgraphs of size at least $A\subset G$6, for any fixed $A\subset G$7 and sufficiently large $A\subset G$8, have edge density close to the expected value (Konyagin et al., 2017).

In the model case $A\subset G$9, the threshold improves: it suffices to take

$E=\{(x,y)\in G\times G:\ x+y\in A\},$0

The proof combines large deviation bounds with additive-combinatorial structure theory, especially additive energy, dissociated sets, and refined counting of low-dimensional subsets. The same work also records the earlier obstruction that one cannot uniformly lower the set-size threshold to $E=\{(x,y)\in G\times G:\ x+y\in A\},$1 (Konyagin et al., 2017).

5. Perfect codes, total perfect codes, and regular sets

Cayley sum graphs support an extensive coding-theoretic theory. In a graph $E=\{(x,y)\in G\times G:\ x+y\in A\},$2, a perfect code is an independent set $E=\{(x,y)\in G\times G:\ x+y\in A\},$3 such that every vertex outside $E=\{(x,y)\in G\times G:\ x+y\in A\},$4 is adjacent to exactly one vertex of $E=\{(x,y)\in G\times G:\ x+y\in A\},$5; a total perfect code is a set $E=\{(x,y)\in G\times G:\ x+y\in A\},$6 such that every vertex of $E=\{(x,y)\in G\times G:\ x+y\in A\},$7 is adjacent to exactly one vertex of $E=\{(x,y)\in G\times G:\ x+y\in A\},$8. In Cayley sum graphs, these notions admit algebraic characterizations (Zhang, 2022, Ma et al., 2020).

For a finite abelian group $E=\{(x,y)\in G\times G:\ x+y\in A\},$9, a square-free subset $x$00, and $x$01, a subset $x$02 is a perfect code in $x$03 if and only if the sets

$x$04

form a partition of $x$05. Equivalently,

$x$06

If $x$07 is inverse-closed, this becomes the direct-sum condition

$x$08

so perfect codes are tilings by a supplementary set (Ma et al., 2020).

Subgroup perfect codes can also be classified. If

$x$09

is a finite abelian group of even order and $x$10 denotes its subgroup of squares, then a subgroup $x$11 is a subgroup perfect code if and only if either $x$12 or $x$13 contains a non-square element (Ma et al., 2020). In a broader nonabelian setting, a subgroup $x$14 is a perfect code in some $x$15 if and only if there exists a normal subset $x$16 such that $x$17 is a left transversal of $x$18 in $x$19; $x$20 is a total perfect code in some $x$21 if and only if there exists a normal left transversal $x$22 whose unique point in $x$23 is a nonsquare element of $x$24 (Zhang, 2022).

Regular-set theory generalizes these notions. A subset $x$25 is $x$26-regular if vertices in $x$27 have exactly $x$28 neighbors in $x$29, while vertices outside $x$30 have exactly $x$31 neighbors in $x$32. For finite abelian groups, if $x$33 and $x$34 are the square and nonsquare elements and

$x$35

then a subgroup $x$36 is an $x$37-regular set of $x$38 if and only if

$x$39

Moreover, $x$40 when $x$41, and otherwise $x$42 (Seiedali et al., 2024). The special case $x$43 recovers subgroup perfect codes.

6. Group-specific classifications and recent developments

Several papers push the classification problem into specific families of groups. For cyclic groups $x$44, total perfect codes in $x$45 are linked to factorizations of the group. If $x$46 is a subgroup, then

$x$47

In the connected cyclic case, if $x$48, then $x$49 admits a subgroup total perfect code if and only if distinct elements of $x$50 are distinct modulo $x$51; the code is then the unique subgroup $x$52 of order $x$53 (Koohestani et al., 23 Oct 2025).

For generalized dicyclic groups $x$54, the existence of subgroup $x$55-regular sets is completely parameterized. The classification splits between subgroups $x$56 and subgroups of the form $x$57, and the admissible parameter pairs depend explicitly on structural data $x$58, $x$59, $x$60, and the square subgroup $x$61. The results include perfect codes and total perfect codes as special cases (Peng et al., 10 Jul 2025).

At the opposite end of the flexibility spectrum lie the symmetric and alternating groups. If $x$62 is a square-free normal subset and $x$63 is the simple product-form Cayley sum graph, then the only subgroup perfect code of $x$64 is $x$65 itself, and the same statement holds for $x$66. In particular, no proper nontrivial subgroup of $x$67 or $x$68 is a perfect code in any such Cayley sum graph (Shaw et al., 5 Sep 2025).

Earlier classification results already showed how restrictive the nonabelian situation can be. For abelian groups, every subgroup is a perfect code in some Cayley sum graph, and every even-order subgroup is a total perfect code in some Cayley sum graph. By contrast, for dihedral groups and for $x$69, the Cayley sum graphs admitting subgroup perfect or total perfect codes can be completely listed, and only very specific connection sets occur in the connected case (Zhang, 2022).

Taken together, these results indicate that the theory of Cayley sum graphs has bifurcated into two complementary regimes. One regime emphasizes spectral expansion, pseudorandomness, Ramanujan phenomena, and twists. The other emphasizes exact algebraic classification of connection sets supporting perfect codes, total perfect codes, or more general regular sets. The common feature is that adjacency by “sum” or “product-to-a-set” preserves enough group structure to make deep representation-theoretic and combinatorial analysis possible, while departing far enough from ordinary Cayley graphs to produce genuinely new behavior (Biswas et al., 2021, Saha, 2023).