  The graph invariant examined in this paper is the largest eigenvalue of the
adjacency matrix of a graph. Previous work demonstrates the tight relationship
between this invariant, the birth and death rate of a contagion spreading on
the graph, and the trajectory of the contagion over time. We begin by
conducting a simulation confirming this and explore bounds on the birth and
death rate in terms of well-known graph invariants. As a result, the change in
the largest eigenvalue resulting from removal of a vertex in the network is the
best measure of effectiveness of interventions that slow the spread of a
contagion. We define the spread centrality of a vertex $v$ in a graph $G$ as
the difference between the largest eigenvalues of $G$ and $G-v$. While the
spread centrality is a distinct centrality measure and serves as another graph
invariant for distinguishing graphs, we found experimental evidence that
vertices ranked by the spread centrality and those ranked by eigenvector
centrality are strongly correlated. Since eigenvector centrality is easier to
compute than the spread centrality, this justifies using eigenvector centrality
as a measure of spread, especially in large networks with unknown portions. We
also examine two strategies for selecting members of a population to vaccinate.
