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Let \( G \) be a connected graph. In this paper, we introduce the concepts of vertex-to-clique \emph{radius} \( r_1 \), vertex-to-clique \emph{diameter} \( d_1 \), clique-to-vertex \emph{radius} \( r_2 \), clique-to-vertex \emph{diameter} \( d_2 \), clique-to-clique \emph{radius} \( r_3 \), and clique-to-clique \emph{diameter} \( d_3 \) in \( G \). We prove that for any connected graph, \( r_i \leq d_i \leq 2r_i + 1 \) for \( i = 1, 2, 3 \). We also find expressions for \( d_1 \), \( d_2 \), and \( d_3 \) for a tree \( T \) in terms of \( r_1 \), \( r_2 \), and \( r_3 \) respectively, which determine the cardinality of each \( Z_i(T) \), where \( Z_i(T) \) is the vertex-to-clique, the clique-to-vertex, and the clique-to-clique center respectively of \( T \) for \( i = 1, 2, 3 \). If \( G \) is a graph that is not a tree and if \( g(G) \) denotes the girth of the graph, then its relation with each of \( d_1 \), \( d_2 \), and \( d_3 \) is discussed. We also characterize the class of graphs \( G \) such that \( G \) is not a tree, \( d_3 \neq 0 \), and \( g(G) = 2d_3 + 3 \).