Lower Bounds for the Domination Numbers of Connected Graphs without Short Cycles

In this paper, we obtain lower bounds for the domination numbers of connected graphs with girth at least $7$. We show that the domination number of a connected graph with girth at least $7$ is either $1$ or at least $\frac{1}{2}(3+\sqrt{8(m-n)+9})$, where $n$ is the number of vertices in the graph and $m$ is the number of edges in the graph. For graphs with minimum degree $2$ and girth at least $7$, the lower bound can be improved to $\max \{\sqrt{n},\sqrt{\frac{2m}{3}}\}$, where $n$ and $m$ are the numbers of vertices and edges in the graph respectively. In cases where the graph is of minimum degree $2$ and its girth $g$ is at least $12$, the lower bound can be further improved to $\max{{\sqrt{n}, \sqrt{\frac{\lfloor \frac{g}{3} \rfloor-1}{3}m}}}$.