Shuriken Graph Operation

• Shuriken Graph Operation is a unary transformation that builds composite multi-spoked graphs using base graphs and ring-theoretic parameters.
• It leverages idempotent and clean graph constructs from finite rings to derive explicit invariants such as clique, chromatic, independence, and domination numbers.
• The operation is algorithmically implemented with complexity O(n²|E(G)|) and finds applications in combinatorics, chemical graph theory, and algebraic graph analysis.

The shuriken graph operation is a unary graph transformation motivated by the interplay of idempotent and unit elements in finite rings with identity, and is abstracted to operate on arbitrary base graphs. Originating from the ring-theoretic context—specifically via the clean graph and the idempotent graph—this operation constructs a family of composite graphs whose invariants and structural properties are explicitly determined by their base graphs and related ring-theoretic parameters (Djuang et al., 22 Jan 2026).

1. Foundational Graph Constructions

Let $R$ be a finite ring with identity. Two auxiliary graphs arise naturally:

• The idempotent graph $I(R)$ is defined as

$$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$

• The clean graph $Cl_2(R)$ is given by

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$

Edges in $Cl_2(R)$ record either orthogonal idempotents or pairs of units that are inverses of each other.

2. The Shuriken Operation: Formal Definition

Given a simple graph $G = (V(G), E(G))$ and parameters $n, t > 0$ with $n-t$ even, the $(t,n)$-shuriken graph $I(R)$0 is constructed as follows:

1. Form the extension $I(R)$1 by adding an isolated vertex $I(R)$2.
2. Construct $I(R)$3 disjoint copies of $I(R)$4, denoted $I(R)$5. For $I(R)$6, let $I(R)$7 denote its image in $I(R)$8; let $I(R)$9 denote the image of $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$0 in $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$1.
3. The vertex set is

$$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$2

1. The edge set consists of: \begin{align*} E(Shu^t_n(G)) =\ & {u_i v_j : uv \in E(G),\ 1 \le i, j \le n} \ \cup\ & {u_i v_i : u \neq v \in V(G) \cup {z},\ 1 \le i \le t} \ \cup\ & {u_i v_{n+t+1-i} : u \in V(G) \cup {z},\ i \in {t+1, ..., \tfrac{n+t}{2}}} \end{align*} This configuration produces a structure resembling a multi-spoked wheel, with inter-copy and intra-copy joins parameterized by $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$3 (number of "spikes") and $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$4 (number of copies).

3. Stepwise Construction from Ring-Theoretic Data

Building $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$5 in the context of a ring $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$6 proceeds as:

• Identify non-trivial idempotents in $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$7 to construct $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$8.
• Form the clean graph $$I(R) = \left(V = \{e \in R : e^2 = e,\ e \notin \{0,1\}\},\ E = \{\{e, f\} : ef = fe = 0\}\right).$$9 with vertices as pairs of idempotents and units reflecting the algebraic interactions.
• Set $Cl_2(R)$0 as the base graph; often $Cl_2(R)$1.
• Apply the shuriken operation as defined above, with the parameter $Cl_2(R)$2 encoding the ring-theoretic idempotent structure. The resulting graph encodes both local and global interactions present in the ring.

4. Explicit Formulas for Classical Invariants

For $Cl_2(R)$3, major graph invariants admit closed formulas:

• Clique Number

$Cl_2(R)$4

• Chromatic Number (Lower Bound)

$Cl_2(R)$5

where

$Cl_2(R)$6

• Independence Number

$Cl_2(R)$7

• Domination Number

$Cl_2(R)$8

5. Degree-Based Topological Indices

Topological indices crucial for chemical graph theory and combinatorics are expressed as follows, letting $Cl_2(R)$9, $$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$0, $$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$1, $$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$2:

• Vertex Degrees in $$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$3

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$4

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$5

• First Zagreb Index

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$6

• Second Zagreb Index

The closed-form is a polynomial in $$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$7, computed by summing degree products over all six edge types generated in the construction.

6. Eulerian and Hamiltonian Properties

• Hamiltonicity

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$8

The construction allows the forming of a cycle traversing all vertices by alternating through spikes and mirrored pairs.

• Eulerian Criterion

$$Cl_2(R) = \left( V = \{(e, u) : e^2 = e \neq 0,\ u \in R^\times\},\ E = \{\{(e,u),(f,v)\} : ef = fe = 0\ \text{or}\ uv = vu = 1\} \right).$$9

Parities of all vertex degrees must be checked as per the closed formulas above.

7. Detailed Example: $Cl_2(R)$0

Let $Cl_2(R)$1:

• Non-trivial idempotents: $Cl_2(R)$2, thus $Cl_2(R)$3.
• Units: $Cl_2(R)$4.
• $Cl_2(R)$5 has 6 vertices.
• $Cl_2(R)$6, $Cl_2(R)$7. Thus $Cl_2(R)$8.
• $Cl_2(R)$9, $G = (V(G), E(G))$0 by order and size formulas.

Key invariants:

• $G = (V(G), E(G))$1
• $G = (V(G), E(G))$2
• $G = (V(G), E(G))$3
• $G = (V(G), E(G))$4
• $G = (V(G), E(G))$5 implies $G = (V(G), E(G))$6

Structural properties:

• $G = (V(G), E(G))$7 is not Hamiltonian, but admits a Hamiltonian path; thus $G = (V(G), E(G))$8 is Hamiltonian.
• $G = (V(G), E(G))$9, hence $n, t > 0$0 is not Eulerian.

8. Algorithmic Construction and Complexity

Construction involves:

• Input $n, t > 0$1.
• Create $n, t > 0$2 copies of $n, t > 0$3.
• Replicate all edges across copies ($n, t > 0$4 complexity).
• Connect spikes and mirrored pairs ($n, t > 0$5).

The major computational expense is the $n, t > 0$6 step of replicating each base edge into all inter-copy pairs.

---

The shuriken graph operation provides a general framework linking ring-theoretic graph constructions (idempotent and clean graphs) with combinatorial invariants and complex connectivity properties, facilitating direct translation of algebraic data into graph-theoretic structures whose properties are expressed in terms of their base graphs (Djuang et al., 22 Jan 2026).