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 $Shu^t_n(G)$ is constructed as follows:

1. Form the extension $G' = G \cup \{z\}$ by adding an isolated vertex $z$.
2. Construct $n$ disjoint copies of $G'$, denoted $G'_1, \ldots, G'_n$. For $v \in V(G)$, let $v_i$ denote its image in $G'_i$; let $z_i$ denote the image of $z$ in $G'_i$.
3. The vertex set is

$$V(Shu^t_n(G)) = \bigcup_{i=1}^n \left(\{z_i\} \cup \{v_i : v \in V(G)\}\right).$$

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 $t$ (number of "spikes") and $n$ (number of copies).

3. Stepwise Construction from Ring-Theoretic Data

Building $Shu^t_n(G)$ in the context of a ring $R$ proceeds as:

• Identify non-trivial idempotents in $R$ to construct $I(R)$.
• Form the clean graph $Cl_2(R)$ with vertices as pairs of idempotents and units reflecting the algebraic interactions.
• Set $G = Cl_2(R)$ as the base graph; often $t = |V(I(R))|$.
• Apply the shuriken operation as defined above, with the parameter $t$ 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 $H = Shu^t_n(G)$, major graph invariants admit closed formulas:

• Clique Number

$$\omega(H) = \begin{cases} |V(G)| + 1, & n-t = 0, \ \max\{|V(G)| + 1, 2 \omega(G)\}, & n-t > 0. \end{cases}$$

• Chromatic Number (Lower Bound)

$$\chi(H) = |V(G)| + 1 \quad \text{if}\ n-t=0, \qquad \chi(H) \geq \max\{|V(G)|+1, 2\chi(G)+\varphi\} \quad \text{if}\ n-t>0,$$

where

$\varphi = \sum_{1 \le k \le \chi(G), |A_k|>2} (|A_k|-2),\quad A_k = \{x: f(x) = k,\ x\ \text{adjacent to all non-%%%%31%%%%–colored vertices}\}$

• Independence Number

$$\alpha(H) = t + \frac{n-t}{2}\left(\alpha(G) + 1\right)$$

• Domination Number

$$\gamma(H) = \begin{cases} \frac{n+t}{2}, & \gamma(G) \leq t, \ \text{an integer in}\ \left[\frac{n+t}{2}, \gamma(G)+\frac{n-t}{2}\right], & \gamma(G) \geq t \end{cases}$$

5. Degree-Based Topological Indices

Topological indices crucial for chemical graph theory and combinatorics are expressed as follows, letting $v = |V(G)|$, $e = |E(G)|$, $M_1(G) = \sum d_G(x)^2$, $M_2(G) = \sum_{xy \in E(G)} d_G(x)d_G(y)$:

• Vertex Degrees in $H$

$$d_H(x_i) = \begin{cases} d_G(x)\,(n-1) + v, & 1 \le i \le t, \ d_G(x)\,(n-1) + (v+1), & t+1 \le i \le n \end{cases}$$

$$d_H(z_i) = \begin{cases} v, & 1 \le i \le t, \ v+1, & t+1 \le i \le n \end{cases}$$

• First Zagreb Index

$\begin{split} M_1(H) =\;& n(n-1)^2\,M_1(G) + n\,v^3 + (3n - 2t)v^2 + 4(n-t)v + (n-t) \ &+ 4(n-1)e\,(n\,v + (n-t)) \end{split}$

• Second Zagreb Index

The closed-form is a polynomial in $n, t, v, e, M_1(G), M_2(G)$, computed by summing degree products over all six edge types generated in the construction.

6. Eulerian and Hamiltonian Properties

• Hamiltonicity

$\text{If %%%%38%%%% contains a Hamiltonian path, %%%%39%%%% is Hamiltonian for all %%%%40%%%% with %%%%41%%%% even.}$

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

• Eulerian Criterion

$Shu^t_n(G)\ \text{is Eulerian iff}\ t = n,\ v = |V(G)|\ \text{even},\ \text{and either %%%%42%%%% is Eulerian or %%%%43%%%% is odd}.$

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

7. Detailed Example: $\mathbb{Z}/6\mathbb{Z}$

Let $R = \mathbb{Z}_6$:

• Non-trivial idempotents: $\{3,4\}$, thus $|V(I(R))| = 2$.
• Units: $\{1,5\}$.
• $G = Cl_2(\mathbb{Z}_6)$ has 6 vertices.
• $t = 2$, $n = 4$. Thus $H = Shu^2_4(G)$.
• $|V(H)| = 28$, $|E(H)| = 121$ by order and size formulas.

Key invariants:

• $\omega(H) = 7$
• $\chi(H) = 7$
• $\alpha(H) = 8$
• $\gamma(H) = 3$
• $M_1(G) = 72$ implies $M_1(H) = 6914$

Structural properties:

• $G$ is not Hamiltonian, but admits a Hamiltonian path; thus $H$ is Hamiltonian.
• $t \neq n$, hence $H$ is not Eulerian.

8. Algorithmic Construction and Complexity

Construction involves:

• Input $(G, n, t)$.
• Create $n$ copies of $V(G) \cup \{z\}$.
• Replicate all edges across copies ($O(n^2|E(G)|)$ complexity).
• Connect spikes and mirrored pairs ($O(n|V(G)|^2)$).

The major computational expense is the $O(n^2|E(G)|)$ step of replicating each base edge into all inter-copy pairs.

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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).