1 Introduction
2 Main results
Notations | Acronyms |
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Total edge irregularity strength | TEIS |
Square snake graphs | \(C_{4,n}\) |
double square snake graph | \(D\left( {C_{4,n} } \right)\) |
Triple square snake graph | \(T\left( {C_{4,n} } \right)\) |
m-multiple square snake graph | \(M_m \left( {C_{4,n} } \right)\) |
Total edge irregularity strength of a graph \(G\) | \(tes\left( G \right)\) |
The maximum degree of vertices of a graph \(G.\) | \(\Delta G\) |
The number of edges of a graph \(G\) | \(\left| {E(G)} \right|\) |
\(1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3}\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right)\) | I |
\(1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right)\) | II |
\(\frac{\lambda^{\!\!\!\!\!-}}{3} + 1 \le i \le n\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right)\) | III |
\(\frac{\lambda^{\!\!\!\!\!-}}{3} + 2 \le i \le n\quad {\text{if}}\quad \lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right)\) | IV |
\(2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3}\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right)\) | V |
\(2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right)\) | VI |
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I is the condition:$$ 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} \quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right) \quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$
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II is the condition:$$ 1 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$
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III is the condition:$$ \frac{\lambda^{\!\!\!\!\!-}}{3} + 1 \le i \le n\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$
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IV is the condition:$$ \frac{\lambda^{\!\!\!\!\!-}}{3} + 2 \le i \le n\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$
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V is the condition:$$ 2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3}\quad {\text{if}}\, \lambda^{\!\!\!\!\!-} \equiv 0\left( {\bmod\,3} \right)\quad {\text{or}}\quad \lambda^{\!\!\!\!\!-} \equiv 1\left( {\bmod\,3} \right) $$
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VI is the condition:$$ 2 \le i \le \frac{\lambda^{\!\!\!\!\!-}}{3} + 1\quad {\text{if}}\,\lambda^{\!\!\!\!\!-} \equiv 2\left( {\bmod\,3} \right) $$