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Let 
$H$ be a group. The co-maximal graph of subgroups of $H$, denoted by 
$\Gamma \left( H \right)$, is a graph whose vertices are non-trivial and proper subgroups of $H$ and two distinct vertices 
$L$ and 
$K$ are adjacent in $\Gamma \left( H \right)$ if and only if 
$H\,=\,LK$. In this paper, we study the connectivity, diameter, clique number, and vertex chromatic number of $\Gamma \left( H \right)$. For instance, we show that if $\Gamma \left( H \right)$ has no isolated vertex, then $\Gamma \left( H \right)$ is connected with diameter at most 3. Also, we characterize all finitely groups whose co-maximal graphs are connected. Among other results, we show that if $H$ is a finitely generated solvable group and $\Gamma \left( H \right)$ is connected, and moreover, the degree of a maximal subgroup is finite, then $H$ is finite. Furthermore, we show that the degree of each vertex in the co-maximal graph of a general linear group over an algebraically closed field is zero or infinite.
