Measures of spread of information are introduced, with applications to neuronal activity in regions of a brain or to the design of artificial robotic networks in which efficient transmission of information is sought. We make links to spectral connectivity measures in graphs, such as spanning trees and higher-order diameters (as defined here). The exposition is then specialized to regular graphs by developing a formula that expresses the number of spanning trees in terms of walks in the complementary graph. Using traces, we then develop bounds for the number of spanning trees. Two approaches are used to establish such bounds: the first involves a logarithmic series expansion of the number of spanning trees in the complementary graph, while the second relies on certain \(l_p\) norm inequalities. Consequences to bipartite graphs are then examined.