Prime Coprime Graph in Finite Groups

• Prime coprime graph is a graph on a finite group where two distinct elements are adjacent if the gcd of their orders is 1 or a prime.
• This construction extends the classical coprime graph by including prime gcds, thereby revealing intricate connectivity and dominating vertex properties.
• It facilitates comprehensive analyses of group invariants, including planarity, Hamiltonicity, clique numbers, and spectral characteristics across group families.

Searching arXiv for recent and foundational papers on prime-coprime / coprime order graphs of finite groups. The prime-coprime graph of a finite group, usually denoted $\Theta(G)$, is the simple graph with vertex set $G$ in which two distinct elements $x$ and $y$ are adjacent whenever $\gcd(o(x),o(y))$ is either $1$ or a prime. In the literature this object also appears under the names coprime order graph and co-prime order graph. It was introduced as a graph on element orders that strictly extends the classical coprime graph of a group: the older graph uses only the condition $\gcd(o(x),o(y))=1$, whereas $\Theta(G)$ enlarges the edge set by also allowing prime gcds (Banerjee, 2019). Subsequent work has developed its structural theory for finite groups, including completeness, planarity, Hamiltonicity, clique number, vertex degree, splitness, independence number, vertex connectivity, and genus (Ranjan et al., 20 Apr 2026, Ranjan et al., 21 Jul 2025, Ma et al., 2020).

1. Terminology and formal definition

The graph was introduced in 2019 under the name coprime order graph of a finite group $G$, with notation $\Theta(G)$, and with adjacency rule

$G$0

for distinct $G$1 (Banerjee, 2019). Later papers use the label prime-coprime graph for the same construction and retain the notation $G$2 (Ranjan et al., 20 Apr 2026, Ranjan et al., 21 Jul 2025). The 2020 paper on the co-prime order graph uses the same adjacency condition in its abstract (Ma et al., 2020).

The relation to earlier coprimality graphs is explicit. The classical coprime graph of a group is a subgraph of $G$3, because its edge condition is only $G$4 (Banerjee, 2019). Two auxiliary subsets recur in the later literature: $G$5 and

$G$6

These sets isolate the vertices of order $G$7 or prime order, which control much of the global structure (Ranjan et al., 20 Apr 2026, Ranjan et al., 21 Jul 2025).

A basic terminological caution is necessary. In the surrounding literature, “coprime graph” may also refer to graphs on integers, subgroup orders, or conjugacy class sizes. The prime-coprime graph $G$8 is specifically a graph on the elements of a finite group, with adjacency determined by the gcd of element orders.

2. Fundamental properties and complete graphs

Several global properties hold for arbitrary finite groups. The graph $G$9 is connected and satisfies $x$0, because the identity element is adjacent to every vertex (Banerjee, 2019). If the girth is finite, then it is $x$1 (Banerjee, 2019). The construction is functorial with respect to isomorphism in the sense that $x$2 implies $x$3, while the converse need not hold (Banerjee, 2019).

The dominating vertices are completely understood. One paper states that every element of $x$4 is a dominating vertex of $x$5 (Ranjan et al., 20 Apr 2026). Another gives the sharper equivalence: an element $x$6 is a dominating vertex of $x$7 if and only if $x$8 or $x$9 is prime (Ranjan et al., 21 Jul 2025). This characterization explains the persistent role of $y$0 and $y$1 in decomposition theorems.

Completeness is also characterized. The graph $y$2 is complete if and only if $y$3 has no elements of composite order (Banerjee, 2019). Equivalently,

$y$4

where an EPO-group is a group in which every nonidentity element has prime order (Ranjan et al., 21 Jul 2025). This criterion makes the complete case purely group-theoretic.

The Eulerian case is unusually rigid: $y$5 is Eulerian if and only if $y$6 is odd and every non-identity element of $y$7 has prime order (Banerjee, 2019). In this sense, the graph is dense for formal reasons, but strong regularity properties still force restrictive group structure.

3. Splitness, joins, and order-type decompositions

A central structural result concerns split graphs. The 2026 paper on independence numbers proves that

$y$8

if and only if one of the following holds: $y$9; or $\gcd(o(x),o(y))$0 consists exclusively of elements of order $\gcd(o(x),o(y))$1, with $\gcd(o(x),o(y))$2 fixed and $\gcd(o(x),o(y))$3; or $\gcd(o(x),o(y))$4 consists exclusively of elements of order $\gcd(o(x),o(y))$5, with $\gcd(o(x),o(y))$6 fixed primes (Ranjan et al., 20 Apr 2026). The proof uses the forbidden induced subgraphs $\gcd(o(x),o(y))$7, $\gcd(o(x),o(y))$8, and $\gcd(o(x),o(y))$9. An immediate consequence is that

$1$0

(Ranjan et al., 20 Apr 2026).

For major families, the graph admits explicit join decompositions. The following descriptions are stated for cyclic, dihedral, and dicyclic groups (Ranjan et al., 21 Jul 2025):

Group family	Prime-coprime graph structure
$1$1 with $1$2	$1$3
$1$4 with distinct primes $1$5	$1$6
$1$7	$1$8
$1$9 with $\gcd(o(x),o(y))=1$0 odd	$\gcd(o(x),o(y))=1$1

The dihedral decomposition reflects the fact that the cyclic subgroup $\gcd(o(x),o(y))=1$2 contributes $\gcd(o(x),o(y))=1$3, while the $\gcd(o(x),o(y))=1$4 reflections all have order $\gcd(o(x),o(y))=1$5 and therefore form a clique $\gcd(o(x),o(y))=1$6 (Ranjan et al., 21 Jul 2025). Earlier work described this same phenomenon informally as a union of the cyclic part $\gcd(o(x),o(y))=1$7 and a $\gcd(o(x),o(y))=1$8 coming from the reflections $\gcd(o(x),o(y))=1$9 (Banerjee, 2019).

For more complicated cyclic and dihedral orders, the same paper gives $\Theta(G)$0-join descriptions and $\Theta(G)$1-partitions. In particular, $\Theta(G)$2 is a $\Theta(G)$3-graph, $\Theta(G)$4 is a $\Theta(G)$5-graph, and $\Theta(G)$6 is a $\Theta(G)$7-graph; analogous statements are proved for the corresponding dihedral groups (Ranjan et al., 21 Jul 2025). This suggests that the graph is often best understood by partitioning $\Theta(G)$8 according to element-order types and then checking adjacency by gcd constraints between those types.

4. Hamiltonicity, connectivity, planarity, and genus

Hamiltonicity has been worked out in detail for three classical families. For cyclic groups,

$\Theta(G)$9

where $G$0 is an odd prime (Ranjan et al., 21 Jul 2025). Earlier work had already shown that for $G$1 with distinct primes $G$2,

$G$3

(Banerjee, 2019). For dihedral groups, the result is absolute: $G$4 (Ranjan et al., 21 Jul 2025). For dicyclic groups,

$G$5

(Ranjan et al., 21 Jul 2025).

Vertex connectivity is known in several cases. For cyclic groups, if $G$6 is prime then

$G$7

while if $G$8 is composite then

$G$9

In particular,

$\Theta(G)$0

(Banerjee, 2019). The later co-prime order graph paper states in its abstract that it computes the vertex-connectivity of the co-prime order graph of a cyclic group, a dihedral group, and a generalized quaternion group, answering a question by Banerjee (Ma et al., 2020).

Planarity appears in both family-specific and global forms. For cyclic groups,

$\Theta(G)$1

(Banerjee, 2019). The 2020 co-prime order graph paper states in its abstract that it classifies all finite groups whose co-prime order graphs are planar (Ma et al., 2020). The same abstract also states that, for a fixed positive integer $\Theta(G)$2, there are finitely many finite groups whose co-prime order graphs have (non)orientable genus $\Theta(G)$3, with applications classifying all finite groups whose co-prime order graphs have (non)orientable genus one and two (Ma et al., 2020). Since only the abstract is available in the supplied data, these claims identify the scope of the classification but not the explicit group lists.

5. Independence number, clique number, degree, and spectra

The independence theory of $\Theta(G)$4 is developed around semiprime divisors. For composite $\Theta(G)$5, let

$\Theta(G)$6

If $\Theta(G)$7 and $\Theta(G)$8, then $\Theta(G)$9 is a maximal independent set of $G$00, and therefore

$G$01

(Ranjan et al., 20 Apr 2026). This lower bound is the starting point for exact calculations.

For cyclic groups, exact formulas are given in several important cases: $G$02

$G$03

$G$04

and

$G$05

(Ranjan et al., 20 Apr 2026). The same paper notes examples such as $G$06 and $G$07, showing that the general semiprime-divisor lower bound need not be tight (Ranjan et al., 20 Apr 2026).

For dihedral groups,

$G$08

(Ranjan et al., 20 Apr 2026). For dicyclic and semidihedral groups, the exact values are also explicit. If $G$09 is odd and $G$10, then

$G$11

If $G$12 with $G$13 and $G$14 odd, then

$G$15

(Ranjan et al., 20 Apr 2026).

Clique numbers are computed in the 2025 paper. For cyclic groups, the paper states that a maximum clique contains all vertices in $G$16, together with one vertex for each prime square divisor $G$17, and one vertex for each product $G$18 with $G$19 (Ranjan et al., 21 Jul 2025). For dihedral groups,

$G$20

and for dicyclic groups,

$G$21

(Ranjan et al., 21 Jul 2025).

Degree formulas are likewise obtained through decomposition. If $G$22 has composite order in $G$23, then

$G$24

and if $G$25 has composite order $G$26 in $G$27, then

$G$28

(Ranjan et al., 21 Jul 2025). Earlier work also investigated exact degree counts for finite abelian groups and dihedral groups, and studied the Laplacian spectrum of the co-prime order graph for finite abelian $G$29-groups, $G$30, and dihedral groups $G$31 (Sehgal et al., 2020). The 2019 introductory paper computed the signless Laplacian spectrum of $G$32 when $G$33 and $G$34 for $G$35 (Banerjee, 2019).

6. Related constructions and common points of confusion

The prime-coprime graph sits within a broader cluster of arithmetic graph constructions, and several of them are easy to conflate.

The first distinction is with the coprime graph of a finite group usually denoted $G$36, where two distinct elements are adjacent if and only if their orders are relatively prime: $G$37 This graph is strictly smaller than $G$38 because it omits the “prime gcd” edges (Ma et al., 24 Jan 2025, Banerjee, 2019).

A second distinction is with the coprime graph of subgroups $G$39, whose vertices are the proper subgroups of $G$40, with adjacency defined by coprime subgroup orders (Rajkumar et al., 2015). That graph is built from subgroup structure rather than element-order interaction.

A third distinction concerns common divisor graphs. The graph $G$41, also denoted $G$42 in the supplied summary, is defined on the sizes of the $G$43-regular conjugacy classes of a finite $G$44-separable group; two vertices are adjacent when they are not coprime: $G$45 Its main theorem states that if $G$46 is $G$47-regular for some $G$48, then it is a complete graph with exactly $G$49 vertices (Sotomayor, 2024). This is an arithmetic graph on conjugacy class sizes, not on group elements.

Finally, the phrase “coprime graph” is also standard for graphs on integers. For example, $G$50 denotes the coprime graph on $G$51, with adjacency $G$52 (Du et al., 26 May 2026). In graph labeling theory, a prime or minimum coprime labeling of a graph can be interpreted as embedding it into such an integer coprime graph (Lee, 2019). This suggests a conceptual parallel: $G$53 is a group-theoretic graph defined by arithmetic of element orders, whereas the integer coprime graph is an ambient host graph for labeling problems.

Taken together, these distinctions show that the prime-coprime graph is one member of a larger arithmetic-graph family, but one with a particularly direct translation between finite-group structure and graph invariants. The later literature, especially on splitness, Hamiltonicity, clique number, and independence number, indicates that this translation is unusually rigid for cyclic, dihedral, dicyclic, and related families (Ranjan et al., 20 Apr 2026, Ranjan et al., 21 Jul 2025).