Net structure in graphs

A graph in said to be a general (g,h) not if and only if (1) it has girth g and (2) it contains a collection of circuits of length h called special circuits such that, every edge of the graph in in precisely two of the circuits. (h is the mesh). When the graph is trivalent it can be embedded on a surface no that the special circuits bound the faces. The definition can be modified so that thin property holds for arbitrary valency y. Nets which have even Euler characteristic but for which the embedding is non-orientable are called quirks. Trivalent nets are considered and the existence of (1) triangle nets (girth 3) for arbitrary mesh h >,3 (2) quirks (3) bipartite hamiltonian nets (mesh - order (number of vertices)) is settled. The vertex-special circuit incidence matrix is called a net matrix. It is shown that when conditions are imposed on g-nets (mesh girth) interesting matrix equations result and that for trivalent nets the spectrum of the dual is restricted. Trivej.ont g-nets of minimum order for g C 6 are known graphs but it is shown that there is no 7-not of order 28 even though the 7-cage has order 24. Two (6,7) nets are determined however which satisfy strong necessary conditions and are called near 7-nets. If the special circuits in a hamiltonian net degenerate into cycles then the net is called a global net. Trivalent orientable global nets are shown to be bipartite and this motivates the definition of a word net. A word net is determined by a single word relation of even length 2. It is shown that finite word nets are always (g,h) nets and complete classifications are given for (1) length 4 and y e(3,4,5} (2) trivalent graphs with I a{6,8}.