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Additional resources for Combinatorial Mathematics III: Proceedings of the Third Australian Conference Held at the University of Queensland, 16–18 May, 1974

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Yacullo This paper studies the family of cyclic sequences of edges (of a connected plane graph) obtained by walking on edges in such a way that the next edge is, alternately, the one that is leftmost or rightmost with respect to the current edge. 1. INTRODUCTION Let C(G) denote the linear space c o n s i s t i n g of sets of edges of a connected graph respectively authors G under symmetric difference, and Z(G) are disjoint. Z(G) Such matters illuminate the structure of in case they are disjoint, C(G) then, by c o u n t i n g dimensions, e of G.

G of order v. Let f Then the group m a t r i x be a M(G,f) = (Mg,h) is defined by indexing a vxv m a t r i x by the elements of and d e f i n i n g mg,h = G by m g , h = f(h-g). ,d k} is defined by f(d) = 1 then if M(f) d e D is and otherwise. Strictly speaking M(f) is unique only up to a p e r m u t a t i o n of rows and columns but this is in no way r e l e v a n t to the present discussion. Group m a t r i c e s have an i n t e r e s t i n g p r o p e r t y w h i c h we refer to as the invariant scalar product p r o p e r t y (ISP property).

Let R be a finite ring w i t h unit. the additive group of R normal function. J(R) N(R,f) = N(f) N(f) Let A restricted function with the p r o p e r t y that f(1) = i denote the group of units of f on is called a R. Let be defined by = {g:g e J(k); f(gS) Because of the importance of = f(g)f(e) N(f) for every 0 E R}. in the next t h e o r e m we d e m o n s t r a t e a structural p r o p e r t y of this set. Proposition Proof. N(f) Let g and f(ghe) Since R is a subgroup of h be members of = f(g)f(hS) J(R).

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