INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025

On Hybrid Order-Sum Graphs of Finite Dihedral Groups

Ibrahim, M.^1,2, Isah, S. H.²

^1,2Department of Mathematic, Usmanu Danfodio University, Sokoto

²Department of Mathematics, Nasarawa State University Keffi, Nasarawa.

*Corresponding Author

DOI: https://doi.org/10.51244/IJRSI.2025.12110129

Received: 03 December 2025; Accepted: 09 December 2025; Published: 18 December 2025

ABSTRACT

This paper introduces a novel graph construction, the inverse-order sum graph, for the dihedral group 퐷₂ₙ. By

merging the adjacency conditions of the inverse graph and the order sum graph, we define 훤ᵢᵥₒₙ(퐷₂ₙ) and

investigate its fundamental graph properties. For n odd, we establish explicit formulas for vertex degrees, graph

sizes, and completeness. The inverse graph 훤 (퐷₂ₙ) exhibits the highest connectivity with degrees 푛 − 1 in

퐼푣

푃₁ ∪ 푃₃ and 푛 − 2 푖푛 푃₂, while the order sum graph 훤 (퐷₂ₙ) is sparser with edges only between full-order

푂푆

rotations. The inverse-order sum graph 훤_퐼푣푂푆(퐷₂ₙ) is the most restrictive, yielding 푛 − 3 degrees in 푃₃ and

isolated vertices elsewhere. Our comparative analysis reveals strict inclusion relations and structural hierarchies

among these graphs, demonstrating how combining algebraic conditions produces refined graphical

representations of group elements. These results contribute to algebraic graph theory by providing new tools for

analyzing finite group structures through hybrid graph constructions, with potential applications in group-based

cryptography and network modeling.

Keywords: inverse graph, order sum graph, inverse-order sum graph, dihedral group, hybrid algebraic graph;

network modelling.

INTRODUCTION

This paper introduces a new graph construction, the inverse-order sum graph, for the dihedral group 퐷_2ꢀ. Recent

studies have explored various graph-theoretic concepts and their applications in group theory.

Romdhini and Nawawi [1] derived the general formula for the degree sum energy of the non-commuting graph

associated with dihedral groups 퐷_2ꢀof order 2푛 (where 푛 ≥ 3), defined as 퐸퐷ꢁ(훤퐺), with the degree sum

matrix entries given by 푑ꢂ푝 + 푑ꢂ푞.

Meanwhile, Ali et al. [2] established new upper bounds for the inverse sum 푖푛푑푒푔 (ꢃꢁꢃ) index of connected

graphs, expressed as

푑_ꢄ푑_푗

( )

ꢃꢁꢃ 퐺 = ∑

푑_ꢄ+ 푑_푗

ꢄ∼푗

where 푖 ∼ ꢅdenotes adjacency between vertices 푖 and ꢅ.

Further contributions include Naduvath's [3] introduction of the order sum graph of finite groups, analyzing its

structural properties such as chromatic number, domination number, and spectral characteristics. Soto [4]

investigated the irreducible representations of the dihedral group 퐷_2ꢀ, determining that it has 푘 + 2 표푟 푘 + 3

irreducible representations depending on whether n is odd or even. Tizard [5] expanded on domination theory in

order sum graphs, examining variants like connected, global, and secure domination. Qasem et al. [6] studied

ꢇ

푚

the sum graph of 푍_{

groups, extending graph theory applications to algebraic structures and computing

}

ꢆ ꢈ

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

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topological indices. Jamal and Rather [7] analyzed the inverse sum indeg energy of graphs, deriving bounds and

constructing equienergetic graph pairs. Hasani [8] identified extremal molecular graphs with minimal and

maximal ISI values, while Hafeez and Farooq [9] formulated ISI energy for specific graph classes and established

related bounds. Gutman et al. [10] contributed by refining lower bounds for the ISI index through a

comprehensive review of existing literature.

A significant gap exists in the literature, as no graph structure has been developed to combine the properties of

inverse graphs and order sum graphs, despite extensive research on graph structures associated with groups.

Motivated by the combination of two or more properties of a single graph to form a new graph, as studied by

authors such as Magami, Ibrahim, Ashafa, and Gana [11] and Bello, Ali, and Isah [12].

This paper fills that gap by defining the inverse-order sum graph 훤_퐼푂푆(퐺), where two distinct elements are

adjacent if and only if they are inverses and the sum of their orders is at least |G|. We focus on the dihedral group

퐷_2ꢀ(n odd), the smallest non-abelian family containing elements of three distinct orders. The resulting hybrid

graph exhibits dramatically lower density than its parents, offering a refined invariant for symmetry analysis,

Cayley graph comparison, and group-based network modelling (e.g., sensor networks with reflection symmetry

or cryptographic key predistribution exploiting inverse and order constraints).

The remainder of this article is structured as follows: Section 2 outlines the fundamental graph-theoretic and

algebraic concepts required. Subsequent sections present our main results on the properties of Γ_퐼푣푂푆퐺(퐷_2ꢀ)

including vertex degrees, connectivity, and completeness, followed by a conclusion.

Fundamental Graph-Theory and Algebraic Scheme

Fundamental Graph-Theory

This section provides an overview of essential graph-theoretic concepts and previously established results that

applied to achieve our study's main findings. These concepts facilitate understanding of our paper. We focus on

simple graphs, which are undirected and no multiple edges or no loops.

Definition 2.1.1 (Inverse Graph), [16]. The inverse graph of a group G, denoted by Γ (G), is a simple graph

Iv

with vertex set G. Two distinct vertices u and v are adjacent if u ∗ v ∈ T_Γor v ∗ u ∈ T_Γ, where T_Γ= {t ∈

G | t ≠ t − 1}.

Example 2.1.1: The dihedral group D_2n= ⟨r, s ∶ rⁿ= s²= e, srs = r − 1⟩, with n ≥ 3, is an example of

a group with no set of generators whose all elements are non-self-invertible. All non self-invertible elements of

n

the group have the form rⁱ, where i ∈ {1,2, . . . , n} and i =

if n is even. The element s ∈ D_2ncannot be

2

expressed as a product of some finite rⁱor their inverses. Figure 2 shows the inverse graph of group 퐷₆.

Definition 2.1.2 (Order Sum Graph) [3] Order sum Graph of a group 퐺 is a simple graph whose vertices are the

elements of 퐺, and two distinct vertices are adjacent if either the sum of their orders is greater than or equal to

the order of 퐺. The order sum graph of group 퐺 is denoted by Γ_OS(G).

Example 2.1.2: The dihedral group of D_2n= ⟨r, s ∶ rⁿ= s²= e, srs = r − 1⟩, with n ≥ 3, where r³=

2

e, s²= e and rs = sr⁻¹and e = 1, r = r

= 3 and s = rs = r s = 2.

| |

|

Now compute all pairs x ≠ y such that ∣ x ∣ +∣ y ∣ 3 + 3 = 6 ≥ 6: (r, r²).

Definition 2.1.3. (Inverse-order Sum Graph of D_2n), [3]. Let Γ_D

= {t ∈ D_2n|t ≠ t⁻¹}, The inverse-order sum

2n

graph of D_2ndenoted by Γ_IvOSG(D_2n) is a simple graph whose vertices are the elements of D_2n, and two distinct

vertices are adjacent if and only if u ⋅ v ∈ Γ_Dor v ⋅ u ∈ Γ_Dand o(u) + o(v) ≥ |G|, ∀ u ≠ v ∈ D_2n.

2n

Definition 2.1.4 (Complete Graph) [17] A complete graph is a simple graph Γ such that every pair of vertices is

joined by an edge. Any complete graph on n vertices is denoted K_n

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

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Definition 2.1.5 (connected Graph), [14]. A graph Γ is said to be connected if any two distinct vertices of Γ

are joined by a path. Γ is said to be disconnected if Γ is not connected,

Definition 2.1.6. (Self Inverse Elements), [14]. Let (X,∗) be a finite Abelian group, an element 푥 ∈ 푋 is called

self inverse element if x⁻¹= x. The set of all self inverse elements in X is denoted by S_ivn

Definition 2.1.7. (Mutual Inverse Elements). [14] Let (X,∗ ) be a finite Abelian group, an element y ∈ X is

called mutual inverse element if ∃y′ ∈ X with y ≠ y′ such that y⁻¹= y′. The set of all mutual inverse

elements in X is denoted by M_ivn(Sowaity et al., 2020).

Algebraic Schemes

Lemma 2.2.1 Let 퐷_2ꢀ= ⟨푟, 푠⟩ be the dihedral group of order 2푛, where 푛 ≥ 3 is a positive integer (even or

odd). Then the elements of 퐷_2ꢀcan be partitioned into three disjoint subsets based on element orders and

corresponding cardnalities as follows:

푃₁ = {푒}: Identity element of order 1.

푃₂: Elements of order 2 (reflections), with |푃₂| = 푛.

푃₃: Non-identity elements of order n (rotations), with |푃₃| = 푛 − 1.

Proof

Let 퐷_2ꢀ= ⟨푟, 푠⟩ be defined by the relations 푟^ꢀ= 푒, 푠²= 푒, and 푠푟푠 = 푟⁻¹. The group consists of 푛

rotations {푟⁰, 푟¹, … , 푟^ꢀ−1} and n reflections {푠, 푠푟, 푠푟², … , 푠푟^ꢀ−1}, giving a total of 2푛 elements.

Disjoint Subsets:

We define the following three subsets:

푃₁= {푒}, 푃₂= {푟^ꢉ∣ 1 ≤ 푘 ≤ 푛 − 1}, and 푃₃= {푠푟^ꢉ∣ 0 ≤ 푘 ≤ 푛 − 1}.

Clearly, these subsets are pairwise disjoint. The identity element 푒 = 푟⁰appears only in 푃₁. The elements 푟^ꢉ

∈

푃₂are proper rotations and are distinct from both the identity and all reflections. The elements of 푃₃contain the

generator 푠, and thus cannot appear in 푃₁or 푃₂. Therefore, 푃₁∩ 푃₂= ∅, 푃₁∩ 푃₃= ∅, and 푃₂∩ 푃₃= ∅.

Since

푃₁∪ 푃₂∪ 푃₃= {푟^ꢉ∣ 0 ≤ 푘 ≤ 푛 − 1} ∪ {푠푟^ꢉ∣ 0 ≤ 푘 ≤ 푛 − 1} = 퐷_2ꢀ,

we conclude that {푃₁, 푃₂, 푃₃} is a disjoint partition of the group.

Cardinalities:

|

The subset 푃₁consists of only the identity element, so 푃₁= 1. The set 푃₂contains all non-identity rotations,

with 푘 ranging from 1 to 푛 − 1, so 푃₂= 푛 − 1. The set 푃₃contains all reflections 푠푟^ꢉ, with 푘 ranging from

|

0 to 푛 − 1, giving exactly 푛 elements. Hence, the cardinalities are:

|

푃₁= 1, 푃₂= 푛 − 1, 푃₃= 푛.

Lemma 2.2.2

Let 퐷_2ꢀ= ⟨푟, 푠⟩ be the dihedral group of order 2푛, where n ≥ 3 is a positive integer (even or odd) Let the

elements of 퐷_2ꢀbe partitioned into the disjoint subsets 푃₁= {푒}, 푃₂= {푟^ꢉ∣ 1 ≤ 푘 ≤ 푛 − 1}, and 푃₃=

{푠푟^ꢉ∣ 0 ≤ 푘 ≤ 푛 − 1} as defined in the preceding lemma, then for any 푥 ∈ 퐷_2ꢀthe order in these subsets

are as follows:

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

ISSN No. 2321-2705 | DOI: 10.51244/IJRSI |Volume XII Issue XI November 2025

1,

If 푥 ∈ 푃₁,

푛

, where 푥 = 푟^ꢉ, If 푥 ∈ 푃₂,

( )

표 푥 = {

푔푐푑(푘, 푛)

2,

for all 푥 = 푠푟^ꢉIf 푥 ∈ 푃₃

Proof

Let 푥 ∈ 퐷_2ꢀ. We consider three cases based on the partition of 퐷_2ꢀ:

Case 1: 푥 ∈ 푃₁

Then 푥 = 푒, the identity element. By definition of the identity, 푒¹= 푒, and for no smaller positive integer.

Therefore, 표(푒) = 1.

Case 2: 푥 ∈ 푃₂

Then 푥 = 푟^ꢉ, for some 1 ≤ 푘 ≤ 푛 − 1. Since 푟 generates a cyclic subgroup of order 푛, we know that the

ꢀ

ꢉ

(

)

order of 푟 in a cyclic group of order 푛 is given by 표 푟

=

.

ꢊꢋꢌ(ꢉ,ꢀ)

Hence, each 푟^ꢉ∈ 푃₂has order dividing 푛, and its specific value depends on the value of 푘 relative to 푛.

Case 3: 푥 ∈ 푃₃

Then 푥 = 푠푟^ꢉfor some 0 ≤ 푘 ≤ 푛 − 1. We compute the square of 푥 as follows:

ꢉ 2

ꢉ

= 푠푟^ꢉ⋅ 푠푟^ꢉ= 푠 푟 푠 푟

= 푠 푠푟 − 푘 푟

= 푠²푟^−ꢉ. 푟^ꢉ= 푒 ⋅ 푒 = 푒.

(

)

(

)

(

)

푠푟

Since 푥²= 푒, the minimal positive integer m such that 푥^ꢍ= 푒 푖푠 ꢎ = 2. Thus, 표(푥) = 2.

RESULTS AND DISCUSSION

This section establishes theorems for vertex degrees, graph sizes, and completeness, followed by a comparative

analysis with concrete examples.

Theorem 3.1 (Vertex Degree of 휞_푰풗(푫₂ₙ) for n Odd)

For 훤 (퐷₂ₙ) with n odd, the vertex degrees are:

퐼푣

푛 − 1 푓표푟 ꢂ ∈ 푃₁∪ 푃₃

( )

{

If 푛 is odd, deg ꢂ =

푛 − 2 푓표푟 ꢂ ∈ 푃₂

Proof

For ꢂ ∈ 푃₁: The identity is adjacent to all elements in 푇_ꢏ= 푃₃, so 푑푒푔(ꢂ) = |푃₃| = 푛 − 1.

For ꢂ ∈ 푃₂: Any two distinct reflections have product in 푃₃ ⊆ 푇_ꢏ, so each reflection is adjacent to all

other 푛 − 1 reflections.

For ꢂ ∈ 푃₃: A non-identity rotation is adjacent to the identity and to all other non-identity rotations except its

inverse, giving 1 + (푛 − 2) − 1 = 푛 − 2 neighbors

Theorem 3.2 (Size of 휞_푰풗(푫₂ₙ) for n Odd)

(

)(

)

ꢀ−1 2ꢀ−1

| |

The number of edges is 퐸 =

2

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

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Proof

By the Handshaking Lemma and Theorem 4.2.1: Sum of degrees = 1 · (푛 − 1) + 푛 · (푛 − 1) + (푛 − 1) ·

(푛 − 2) = (푛 − 1)(2푛 − 1)

(

)(

)

ꢀ−1 2ꢀ−1

| |

Thus 퐸 =

.

2

Theorem 3.3 (Completeness of 휞_푰풗(푫₂ₙ) for 풏 Odd)

훤 (퐷₂ₙ) is not complete.

퐼푣

Proof

Vertices in 푃₁ and 푃₂ are not adjacent to each other, as their product is not in 푇_ꢏ.

Example

for 푛 = 3 퐷₆ = {푒, 푟, 푟², 푠, 푠푟, 푠푟²}, 푇_ꢏ= {푟, 푟²}.

The edges are only 푒 ∼ 푟, 푒 ∼ 푟², 푠 ∼ 푠푟, 푠 ∼ 푠푟², 푠푟 ∼ 푠푟² (5 edges).

(

)(

)

3−1 6−1

| |

Degrees match the theorem, and 퐸 =

= 5.

2

Theorem 3,4 (Vertex Degree of 휞_푶푺(푫₂ₙ) for n Odd)

0 푓표푟 ꢂ ∈ 푃₁∪ 푃₃

( )

deg ꢂ =

{

푛 − 2 푓표푟 ꢂ ∈ 푃₂

Proof

The condition |푥| + |푦| ≥ 2푛 holds only when both 푥 and 푦 have order 푛, i.e., both are in 푃₃. Thus only

vertices in 푃₃ have neighbors, and each is adjacent to all other 푛 − 2 vertices in 푃₃.

Theorem 3.5 (Size of 휞_푶푺(푫₂ₙ) for n Odd)

(

)(

)

ꢀ−1 ꢀ−2

| |

퐸 =

2

Proof

The graph is the complete graph on 푃₃ (which has 푛 − 1 vertices).

Theorem 3.6 (Completeness of 휞_푶푺(푫₂ₙ) for 풏 Odd)

훤ₒₙ(퐷₂ₙ) is not complete (vertices in 푃₁ ∪ 푃₂ are isolated).

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

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Example

for 푛 = 3 Only the pair (푟, 푟²) satisfies the order-sum condition, giving one edge 푟 ∼ 푟² and |퐸| = 1, as

predicted.

Theorem 2.7 (Vertex Degree of 휞_푰풗푶푺(푫₂ₙ) for n Odd)

0 푓표푟 ꢂ ∈ 푃₁∪ 푃₃

( )

deg ꢂ =

{

푛 − 3 푓표푟 ꢂ ∈ 푃₂

Proof

Adjacency is possible only inside 푃₃ (order-sum condition). Within P₃, the inverse condition additionally forbids

an edge to the unique inverse, so each vertex loses one more neighbor compared to 훤 , yielding 푛 − 3 neighbors.

푂푆

(

)(

)

ꢀ−1 ꢀ−3

| |

Theorem 3.8 (Size of 휞_푰풗푶푺(푫₂ₙ) for 풏 Odd) 퐸 =

2

Proof

(

)(

)

ꢀ−1 ꢀ−2

ꢀ−1

Start from the complete graph on 푃₃ (which has

edges) and remove the

edges corresponding to

2

(

)(

ꢀ−1 ꢀ−3

each rotation and its inverse, giving

⁾edges.

2

Theorem 3.9 (Completeness of 휞_푰풗푶푺(푫₂ₙ) for 풏 Odd)

_퐼푣푂푆(퐷₂ₙ) is not complete (isolated vertices in 푃₁ ∪ 푃₂ and missing inverse edges in 푃₃).

훤

Example for 풏 = ퟑ The only candidate pair (푟, 푟²) fails the inverse condition (푟 · 푟² = 푒 ∉ 푇_ꢏ), so the graph

has no edges (|퐸| = 0), matching the formula.

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INTERNATIONAL JOURNAL OF RESEARCH AND SCIENTIFIC INNOVATION (IJRSI)

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Table 3.1: Comparative Analysis for n Odd

Graph Property

Order

Sum

Graph

Inverse-Order

Sum

Graph

Inverse Graph (휞_푰풗)

(휞_푶푺

)

(휞_푰풗푶푺)

0

Vertex degree in 푃₁

Vertex degree in 푃₂

Vertex degree in 푃₃

Number of edges |퐸|

푛 − 1

푛 − 2

푛 − 3

(

)(

)

(

)(

)

(

)(

)

푛 − 1 2푛 − 1

푛 − 1 푛 − 2

푛 − 2 푛 − 4

2

Completeness

Not complete

DISCUSSION

The Inverse Graph exhibits high connectivity due to its purely algebraic (inverse) adjacency rule, creating edges

across multiple partitions. The Order Sum Graph is considerably sparser, with edges restricted to high-order

elements only. The Inverse-Order Sum Graph is the most restrictive, combining both conditions and producing

the sparsest structure among the three. These differences illustrate how the choice of adjacency criteria

dramatically affects connectivity, with potential applications in algebraic graph theory, group-based network

modeling, and the study of Cayley-type graphs on dihedral groups.

Applications and Implications

The strict hierarchy 훤

퐼푣

⊃ 훤

푂푆

⊃ 훤 provides a tunable family of Cayley-like graphs on the same vertex set,

퐼푂푆

allowing precise control of edge density via algebraic constraints that is useful in symmetry-constrained network

design and key predistribution in group-based cryptography. The almost-complete structure of 훤 on rotations

퐼푂푆

suggests applications in robust synchronisation protocols for systems with reflection symmetry.

CONCLUSION

This research successfully developed and analyzed the inverse-order sum graph 훤ꢃꢂꢐꢁ(퐷₂ₙ) for the dihedral

group D₂ₙ with n odd by merging the adjacency conditions of the inverse graph and the order sum graph,

(

)(

⁾edges, 훤 (퐷₂ₙ) has

ꢀ−1 2ꢀ−1

revealing a hierarchy in graph density: 훤ꢃᵥ(퐷₂ₙ) is the most connected with

푂푆

2

(

)(

)

)(

edges, and 훤ꢃᵥꢐꢁ(퐷₂ₙ) is the sparsest with⁽

⁾edges.

ꢀ−1 ꢀ−2

ꢀ−1 ꢀ−3

2

By merging inverse and order-sum relations, the inverse-order sum graph 훤_퐼푂푆(퐷_2ꢀ) (n odd) produces the

sparsest non-trivial hybrid in the family while preserving algebraic significance, demonstrate the power of hybrid

constructions for refining graphical invariants of non-abelian groups. The results open natural extensions to n

even, dicyclic groups, and spectral or domination parameters.

Author contributions: Conceptualization, Ibrahim, M., Isah, S. H.; Methodology, Ibrahim, M., Isah, S. H.;

Software, Isah, S. H.; Validation, Ibrahim, M.; Formal Analysis, Ibrahim, M.; Investigation, Isah, S. H.; Writing

– original draft preparation, Isah, S. H.; Writing – review and editing, Ibrahim, M.; Visualization, Isah, S. H.;

Supervision, Ibrahim, M.; Project Administration, Ibrahim, M. All authors have read and agreed to the published

version of the manuscript.

Funding Statement: This research received no external funding. Data Availability: Not applicable.

Acknowledgments: The authors thank the anonymous reviewers for their insightful suggestions.

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Conflict of interest: The authors declare no conflict of interest.

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