In this thesis, first, we study the extention of the standard network centrality measures of degree, closeness and
betweenness to apply to groups and classes as well as individuals. The group centrality
measures will enable researchers to answer such questions as 'among middle
managers in a given organization, which are more central, the men or the women?' With
these measures we can also solve the inverse problem: given the network of ties among
organization members, how can we form a team that is maximally central? The measures
are illustrated using two classic network data sets. We also formalize a measure of group
centrality efficiency, which indicates the extent to which a group's centrality is principally due to a small subset of its members. Second, we study the notion of group centralization with respect to eccentricity, degree and betweenness centrality measures. For groups
of size 2, we determine the maximum achieved value of group eccentricity and group betweenness centralization and describe the corresponding extremal graphs. For group degree centralization, we do the same with arbitrary size of group. %We conclude with posing few open problems. Finally, we focus on degree-based measures group degree centrality $GD$ and its centralization $GD_1$. We address the following questions: i) For a fixed $k$, which $k$-subset $S$ of vertices $G$, $\dbinom{V(G)}{k}$, represents the most central group? ii) Among all possible values of $k$, which is the one for which the corresponding set $S$ is most central?
How can we efficiently compute both $k$ and $S$? To answer these questions, we relate with the well-studied
areas of domination and set covers. Using this, we first observe that determining $S$ from the first question
is NP-hard. Then, we describe a greedy approximation algorithm which computes centrality values
over all group sizes $k$ from 1 to $n$ in linear time, and achieve a group degree centrality value of at least
$(1-1/e)(w^\ast - k)$, where $w^\ast$ is