# Some theorems in topology and a possible biological implication

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## Abstract

In a previous paper (*Bull. Math. Biophysics*,**16**, 317–48, 1954) a transformation*T* of one graph into another was suggested, which may describe the relations between organisms of different complexity. In this paper some topological properties of the transformation*T* are studied. It is shown that the fundamental group of the transformed graph is homomorph to the fundamental group of the original graph. An expression is derived for the number of points in a point base of the transformed graph in terms of the number of points of the point base of the original when the point base of the latter consists only of residual points, and it is shown that the ratio of the number of points of the point base to the total number of points of the graph is in that case greater in the transformed graph than in the original. A combinatorial problem arising in connection with the transformation*T* is solved by deriving the number of possible ways in which*n-n* _{ i } indistinguishable elements may be arranged in*n* _{ i } classes, permitting some of the*n* _{ i } classes to be empty.

The possible biological meaning of the increased ratio of the number of points of the point base to the total number of points of the graph is discussed. It is suggested that it may be interpreted as a decrease of regenerating ability with increase of differentiation of the organism. Those considerations suggest the possibility of deriving some general biological laws from the consideration of the properties of the transformation only, regardless of the choice of the primordial graph.

## Keywords

Point Base Fundamental Group Identity Element Representative Point Closed Path## Preview

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## Literature

- König, Denes. 1936.
*Theorie der Endlichen und Unendlichen Graphen*. Leipzig: Akademische Verlagsgesellschaft.Google Scholar - Netto, Eugen. 1901.
*Lehrbuch der Kombinatorik*. Leipzig: B. G. Teubner.MATHGoogle Scholar - Pontrjagin, Leon. 1946.
*Topological Groups*. Princeton: Princeton University Press.Google Scholar - Rashevsky, N. 1954. “Topology and Life: In Search of General Mathematical Principles in Biology and Sociology.”
*Bull. Math. Biophysics*,**16**, 317–48.MathSciNetGoogle Scholar - Seifert, H. and W. Threlfall. 1934.
*Lehrbuch der Topologie*. Leipzig-Berlin: B. G. Teubner.MATHGoogle Scholar