  The co-maximal subgroup graph $\Gamma(G)$ of a group $G$ is a graph whose
vertices are non-trivial proper subgroups of $G$ and two vertices $H$ and $K$
are adjacent if $HK=G$. In this paper, we continue the study of $\Gamma(G)$,
especially when $\Gamma(G)$ has isolated vertices. We define a new graph
$\Gamma^*(G)$, which is obtained by removing isolated vertices from
$\Gamma(G)$. We characterize when $\Gamma^*(G)$ is connected, a complete graph,
star graph, has an universal vertex etc. We also find various graph parameters
like diameter, girth, bipartiteness etc. in terms of properties of $G$.
