  In this paper we give lower bounds on the size of $(a,b)$ elementary trapping
sets (ETSs) belonging to variable-regular LDPC codes with any girth, $g$, and
irregular ones with girth 8, where $a$ is the size, $b$ is the number of
degree-one check nodes and satisfy the inequality $\frac{b}{a}<1$. Our proposed
lower bounds are analytical, rather than exhaustive search-based, and based on
graph theories. The numerical results in the literarture for $g=6,8$ for
variable-regular LDPC codes match our results. Some of our investigations are
independent of the girth and rely on the variables $a$, $b$ and $\gamma$, the
column weight value, only. We prove that for an ETS belonging to a
variable-regular LDPC code with girth 8 we have $a\geq2\gamma-1$ and
$b\geq\gamma$. We demonstrate that these lower bounds are tight, making use of
them we provide a method to achieve the minimum size of ETSs belonging to
irregular LDPC codes with girth 8 specially those whose column weight values
are a subset of $\{2,3,4,5,6\}$. Moreover, we show for variable-regular LDPC
codes with girth 10, $a\geq(\gamma-1)^2+1$. And for $\gamma=3,4$ we obtain
$a\geq7$ and $a\geq12$, respectively. Finally, for variable-regular LDPC codes
with girths $g=2(2k+1)$ and $g=2(2k+2)$ we obtain $a\geq(\gamma-2)^k+1$ and
$a\geq2(\gamma-2)^k+1$, respectively.
