  We introduce a new matrix function for studying graphs and real-world
networks based on a double-factorial penalization of walks between nodes in a
graph. This new matrix function is based on the matrix error function. We find
a very good approximation of this function using a matrix hyperbolic tangent
function. We derive a communicability function, a subgraph centrality and a
double-factorial Estrada index based on this new matrix function. We obtain
upper and lower bounds for the double-factorial Estrada index of graphs,
showing that they are similar to those of the single-factorial Estrada index.
We then compare these indices with the single-factorial one for simple graphs
and real-world networks. We conclude that for networks containing chordless
cycles---holes---the two penalization schemes produce significantly different
results. In particular, we study two series of real-world networks representing
urban street networks, and protein residue networks. We observe that the
subgraph centrality based on both indices produce significantly different
ranking of the nodes. The use of the double factorial penalization of walks
opens new possibilities for studying important structural properties of
real-world networks where long-walks play a fundamental role, such as the cases
of networks containing chordless cycles.
