By Vasyl Ustimenko

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**Extra resources for Algebraic graphs and security of digital communications**

**Example text**

Pi,i , p′i,i , pi,i+1 , pi+1,i , . ), ′ ′ [l] = [l1,0 , l1,1 , l1,2 , l2,1 , l2,2 , l2,2 , l2,3 , . . , li,i , li,i , li,i+1 , li+1,i , . ]. The elements of P and L can be thought as inﬁnite ordered tuples of elements from K, such that only ﬁnite number of components are diﬀerent from zero. We now deﬁne an incidence structure (P, L, I) as follows. We say the point (p) is incident with the line [l], and we write (p)I[l], if the relations (2. 3) between their co-ordinates hold: For each positive integer k ≥ 2 we obtain an incidence structure (Pk , Lk , Ik ) as follows.

Graphs with special walks, definitions and motivations require non polynomial expression f (k, d) for the number of steps (natural branching process give us k(k − 1)(d−) steps). If the distance d is unknown the problem getting harder, the complexity f (k, d) is growing, when d is increasing. One of the popular mathematical models of the procedure for sending a message is the following: (1) treat the information, to be sent, as a vector x = (x1 , . . ) (4) our receiver detects a message y ′ , which may be diﬀerent from y, due to transmission errors.

24 27 32 34 36 40 42 48 24 2. 1. 1. Walks on simple graphs and cryptography A combinatorial method of encryption with a certain similarity to the classical scheme of linear coding has been suggested in [107]. The general idea is to treat vertices of a graph as messages and arcs of a certain length as encryption tools. e. to guess the encoding arc. In fact the quality is good for graphs which are close to the Erd¨os bound, deﬁned by the Even Cycle Theorem. In the case of parallelotopic graphs there is a uniform way to match arcs with strings in the certain alphabet.