  Determining important vertices in large graphs (e.g., Google's PageRank in
the case of the graph of the World Wide Web) facilitated the construction of
excellent web search engines, returning the most important hits corresponding
to the submitted user queries. Interestingly, finding important edges --
instead of vertices -- in large graphs has received much less attention until
now. Here we examine the human structural braingraph (or connectome),
identified by diffusion magnetic resonance imaging (dMRI) methods, with edges
connecting cortical and subcortical gray matter areas and weighted by fiber
strengths, measured by the number of the discovered fiber tracts along the
edge. We identify several "single" important edges in these braingraphs, whose
high or low weights imply the sex or the age of the subject observed. We call
these edges implicator edges since solely from their weight, one can infer the
sex of the subject with more than 67 \% accuracy or their age group with more
than 62\% accuracy. We argue that these brain connections are the most
important ones characterizing the sex or the age of the subjects. Surprisingly,
the edges implying the male sex are mostly located in the anterior parts of the
brain, while those implying the female sex are mostly in the posterior regions.
Additionally, most of the inter-hemispheric implicator edges are male ones,
while the intra-hemispheric ones are predominantly female edges. Our pioneering
method for finding the sex- or age implicator edges can also be applied for
characterizing other biological and medical properties, including
neurodegenerative- and psychiatric diseases besides the sex or the age of the
subject, if large and high-quality neuroimaging datasets become available.
