**4. Clique Number and Girth of** *TI***(Γ(***R***))**

In this section, we study the clique number and girth of *TI*(Γ(R)).

**Theorem 10.** *Let I be a prime ideal of* R *and* |*I*| = *m. Then, TI*(Γ(R)) *has cliques of the form K*1*, K*2*, K*3*, ..., Km*+1*. Moreover, ω*(*TI*(Γ(R))) = *m* + 1*.*

**Proof.** Suppose that *I* is a prime ideal of R and |*I*| = *m*. Then, by using Theorem 3, *TI*(Γ(R)) contains a complete subgraph of order |*I*| +1, and this order is the greatest integer *n* = |*I*| + 1 ≥ 2 such that *Kn* ⊆ *TI*(Γ(R)). Hence, *ω*(*TI*(Γ(R))) = |*I*| + 1 = *m* + 1.

**Theorem 11.** *Let I be an ideal of* R*, which is not prime and there exists u* ∈ *Y such that u*<sup>2</sup> ∈ *I. Then, ω*(*TI*(Γ(R))) ≥ 2|*I*|*.*

**Proof.** Suppose that *I* is not a prime ideal of R. Then, by using Theorem 4, *TI*(Γ(R)) contains a complete subgraph of order at least 2|*I*|. Hence, *ω*(*TI*(Γ(R))) ≥ 2|*I*|.

**Theorem 12.** *Let I be an ideal of* R*, which is not prime, and u*<sup>2</sup> ∈/ *I for all u* ∈ *Y. Then, ω*(*TI*(Γ(R))) ≥ |*I*| + 1*.*

**Proof.** Suppose that *I* is not a prime ideal of R. Then, by using Theorem 5, *TI*(Γ(R)) contains a complete subgraph of order at least |*I*| + 1. Hence, *ω*(*TI*(Γ(R))) ≥ |*I*| + 1.

**Corollary 10.** *Let I be a non-prime ideal of* R *and* |*I*| = *n. Then, ω*(*TI*(Γ(R))) = *ω*(Γ*I*(R)) + *n.*

**Remark 6.** *Let I be an ideal of a ring* R*. Then, ω*(Γ(R/*I*)) ≤ *ω*(Γ*I*(R)) ≤ *ω*(*TI*(Γ(R)))*. Moreover, we know that if* Γ*I*(R) *has no connected columns (i.e., if u*<sup>2</sup> ∈/ *I for all u* ∈ *Y), then ω*(Γ(R/*I*)) = *ω*(Γ*I*(R)) *(for reference see Theorem 4.5 [2]).*

**Theorem 13.** *Let I be a nonzero ideal of a ring* R*. If* |R| ≥ 3*, then TI*(Γ(R)) *has a cycle. Moreover, gr*(*TI*(Γ(R))) = 3*.*

**Proof.** Since *I* is a nonzero ideal, *I* has at least two elements (say *u*, *v*). Moreover, each element of R is adjacent to the elements of *I* and |R| ≥ 3, i.e., there exists *w* ∈ R such that *u* and *v* are adjacent to *w*. Thus, *u* − *w* − *v* − *u* is a cycle of length three, which is the smallest cycle in *TI*(Γ(R)). Hence, *gr*(*TI*(Γ(R))) = 3.

**Corollary 11.** *Let I be an ideal of a ring* R*. If* |R| ≤ 2*, then gr*(*TI*(Γ(R))) = ∞*.*

**Corollary 12.** *Let I be a zero ideal of a ring* R*. Then,*

$$gr(T\_I(\Gamma(\mathcal{R}))) = \begin{cases} 3 & if \, |Y| \ge 2 \\ \infty & if \, |Y| \le 1. \end{cases}$$

**Remark 7.** *Let I be a non-prime ideal of a ring* R*. Then,*

*gr*(*TI*(Γ(R))) ≤ *gr*(Γ*I*(R)) ≤ *gr*(Γ(R/*I*)).
