wp_admin_bar_site_menu

function wp_admin_bar_site_menu( $wp_admin_bar ) {
	// Don't show for logged out users.
	if ( ! is_user_logged_in() ) {
		return;
	}

	// Show only when the user is a member of this site, or they're a super admin.
	if ( ! is_user_member_of_blog() && ! current_user_can( 'manage_network' ) ) {
		return;
	}

	$blogname = get_bloginfo( 'name' );

	if ( ! $blogname ) {
		$blogname = preg_replace( '#^(https?://)?(www.)?#', '', get_home_url() );
	}

	if ( is_network_admin() ) {
		/* translators: %s: Site title. */
		$blogname = sprintf( __( 'Network Admin: %s' ), esc_html( get_network()->site_name ) );
	} elseif ( is_user_admin() ) {
		/* translators: %s: Site title. */
		$blogname = sprintf( __( 'User Dashboard: %s' ), esc_html( get_network()->site_name ) );
	}

	$title = wp_html_excerpt( $blogname, 40, '…' );

	$wp_admin_bar->add_node(
		array(
			'id'    => 'site-name',
			'title' => $title,
			'href'  => ( is_admin() || ! current_user_can( 'read' ) ) ? home_url( '/' ) : admin_url(),
		)
	);

	// Create submenu items.

	if ( is_admin() ) {
		// Add an option to visit the site.
		$wp_admin_bar->add_node(
			array(
				'parent' => 'site-name',
				'id'     => 'view-site',
				'title'  => __( 'Visit Site' ),
				'href'   => home_url( '/' ),
			)
		);

		if ( is_blog_admin() && is_multisite() && current_user_can( 'manage_sites' ) ) {
			$wp_admin_bar->add_node(
				array(
					'parent' => 'site-name',
					'id'     => 'edit-site',
					'title'  => __( 'Edit Site' ),
					'href'   => network_admin_url( 'site-info.php?id=' . get_current_blog_id() ),
				)
			);
		}
	} elseif ( current_user_can( 'read' ) ) {
		// We're on the front end, link to the Dashboard.
		$wp_admin_bar->add_node(
			array(
				'parent' => 'site-name',
				'id'     => 'dashboard',
				'title'  => __( 'Dashboard' ),
				'href'   => admin_url(),
			)
		);

		// Add the appearance submenu items.
		wp_admin_bar_appearance_menu( $wp_admin_bar );
	}
}