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.