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Deep graph-theoretic ideas in the context with the graph of the World WideWeb led to the definition of Google's PageRank and the subsequent rise of themost-popular search engine to date. Brain graphs, or connectomes, are beingwidely explored today. We believe that non-trivial graph theoretic concepts,similarly as it happened in the case of the World Wide web, will lead todiscoveries enlightening the structural and also the functional details of theanimal and human brains. When scientists examine large networks of tens orhundreds of millions of vertices, only fast algorithms can be applied becauseof the size constraints. In the case of diffusion MRI-based structural humanbrain imaging, the effective vertex number of the connectomes, or brain graphsderived from the data is on the scale of several hundred today. That sizefacilitates applying strict mathematical graph algorithms even for somehard-to-compute (or NP-hard) quantities like vertex cover or balanced minimumcut. In the present work we have examined brain graphs, computed from the dataof the Human Connectome Project, recorded from male and female subjects betweenages 22 and 35. Significant differences were found between the male and femalestructural brain graphs: we show that the average female connectome has moreedges, is a better expander graph, has larger minimal bisection width, and hasmore spanning trees than the average male connectome. Since the average femalebrain weights less than the brain of males, these properties show that thefemale brain is more "well-connected" or perhaps, more "efficient" in a sensethan the brain of males.

Graph Theoretical Analysis Reveals: Women's Brains are Better Connected than Men's