Old Theorem

This property tells that G is Kekulean or concealed non—Kekul an. 8.8.2 Old Theorem 

Let S be a set of vertices in G. Then N(S) shall be used to designate the set of vertices in G which are adjacent to the vertices of S, called the neighbour set of S in G. A useful theorem of Hall P (1935) is formulated and supported by an illustration below (cf. also Chen, Cyvin SJ and Cyvin 1990). 

Theorem (Hall P 1935) Let G be a bipartite graph with bipartition (X, Y). Then G has a matching that saturates every vertex in X if and only if N(S) S for all S C X. 

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In this chapter we introduce complex linear algebra, that is, linear algebra where complex numbers are the scalars for scalar multiplication. This may feel like review, even to readers whose experience is limited to real linear algebra. Indeed, most of the theorems of linear algebra remain true if we replace R by C because the axioms for a real vector space involve only addition and multiplication of real numbers, the definition and basic theorems can be easily adapted to any set of scalars where addition and multiplication are defined and reasonably well behaved, and the complex numbers certainly fit the bill. However, the examples are different. Furthermore, there are theorems (such as Proposition 2.11) in complex linear algebra whose analogues over the reals are false. We will recount but not belabor old theorems, concentrating on new ideas and examples. The reader may find proofs in any number of... 

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