Theorems speaking for the asymmetry of all animal brains

In random graph theory it has been proved that with the increasing size of a graph, the proportion of the non-symmetric graphs increases and this class becomes the dominant one while the number of symmetric cases turns to be 'negligible'. Thus the asymmetry (AS) is the generic property. Since nervous systems are representable by graphs or better with special digraphs, the networks, it follows that the brains are asymmetric in a strong sense according to which all cells are distinguishable from each other alone by their internal connections. Such a consequence holds perfectly only if a random evolution or generation of neural networks is supposed. Thus apparent symmetries have to come from heavily controlled (i.e. non random) ontogenetic processes. At the present time the possible total cellular heterogeneity of the various nervous systems has still unclear functional implications. In small nervous systems the odd number of neurons alone is neither a sufficient nor a necessary condition of the asymmetry in the outlined sense.