By Rudenskaya O.G.

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**Extra resources for 4-Quasiperiodic Functions on Graphs and Hypergraphs**

**Example text**

Proof. 9. Then f is well deﬁned, is unique and makes the diagram commutative. We only have to show that %f and f are graph homomorphisms. 8. y//. H /. The two additional properties are the same as for sets, so nothing further needs to be proved. 11. In category-theoretical language, the essence of the Homomorphism Theorem is that every homomorphism has an epi-mono factorization in the given category. Note that in the graph categories considered, epimorphisms (epis) are surjective and monomorphism (monos) are injective.

G/2 without having to know its deﬁnition from linear algebra. The adjacency matrix, the distance matrix and circuits The following remark and two theorems are obvious. 6. If jV j D n, then the length of a simple path in G is at most n. If the length equals n, then the path is a circuit. 7. Let G be a graph with n vertices. r/ (b) dij is the smallest r 2 N with aij > 0 and r < n, if such an r exists; (c) dij D 1 otherwise. r/ (b) rij D 1 if and only if there exists r < n with aij > 0; (c) rij D 0 otherwise.

A similar method works for products of more than two matrices. In all cases, the resulting graph depends on the numbering. G/2 without having to know its deﬁnition from linear algebra. The adjacency matrix, the distance matrix and circuits The following remark and two theorems are obvious. 6. If jV j D n, then the length of a simple path in G is at most n. If the length equals n, then the path is a circuit. 7. Let G be a graph with n vertices. r/ (b) dij is the smallest r 2 N with aij > 0 and r < n, if such an r exists; (c) dij D 1 otherwise.

### 4-Quasiperiodic Functions on Graphs and Hypergraphs by Rudenskaya O.G.

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