the metric dimension and girth of graphs

a set $wsubseteq v(g)$ is called a resolving set for $g$, if for each two distinct vertices $u,vin v(g)$ there exists $win w$ such that $d(u,w)neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. the minimum cardinality of a resolving set for $g$ is called the metric dimension of $g$, and denoted by $dim(g)$. in this paper, it is proved that in a connected graph $g$ of order $n$ which has a cycle, $dim(g)leq n-g(g)+2$, where $g(g)$ is the length of the shortest cycle in $g$, and the equality holds if and only if $g$ is a cycle, a complete graph or a complete bipartite graph $k_{s,t}$, $ s,tgeq 2$.