  Let $box(G)$ be the boxicity of a graph $G$, $G[H_1,H_2,\ldots, H_n]$ be the
$G$-generalized join graph of $n$-pairwise disjoint graphs $H_1,H_2,\ldots,
H_n$, $G^d_k$ be a circular clique graph (where $k\geq 2d$) and $\Gamma(R)$ be
the zero-divisor graph of a commutative ring $R$. In this paper, we prove that
$\chi(G^d_k)\geq box(G^d_k)$, for all $k$ and $d$ with $k\geq 2d$. This
generalizes the results proved in \cite{Aki}. Also we obtain that
$box(G[H_1,H_2,\ldots,H_n])\leq \mathop\sum\limits_{i=1}^nbox(H_i)$. As a
consequence of this result, we obtain a bound for boxicity of zero-divisor
graph of a finite commutative ring with unity. In particular, if $R$ is a
finite commutative non-zero reduced ring with unity, then $\chi(\Gamma(R))\leq
box(\Gamma(R))\leq 2^{\chi(\Gamma(R))}-2$. where $\chi(\Gamma(R))$ is the
chromatic number of $\Gamma(R)$. Moreover, we show that if $N=
\prod\limits_{i=1}^{a}p_i^{2n_i} \prod\limits_{j=1}^{b}q_j^{2m_j+1}$ is a
composite number, where $p_i$'s and $q_j$'s are distinct prime numbers, then
$box(\Gamma(\mathbb{Z}_N))\leq
\big(\mathop\prod\limits_{i=1}^{a}(2n_i+1)\mathop\prod\limits_{j=1}^{b}(2m_j+2)\big)-\big(\mathop\prod\limits_{i=1}^{a}(n_i+1)\mathop\prod\limits_{j=1}^{b}(m_j+1)\big)-1$,
where $\mathbb{Z}_N$ is the ring of integers modulo $N$. Further, we prove
that, $box(\Gamma(\mathbb{Z}_N))=1$ if and only if either $N=p^n$ for some
prime number $p$ and some positive integer $n\geq 2$ or $N=2p$ for some odd
prime number $p$.
