# Algebraic graph theory. Morphisms, monoids and matrices by Ulrich Knauer

By Ulrich Knauer

Graph types are super beneficial for the majority functions and applicators as they play an incredible function as structuring instruments. they enable to version web buildings - like roads, pcs, phones - situations of summary facts constructions - like lists, stacks, timber - and sensible or item orientated programming. In flip, graphs are types for mathematical items, like different types and functors.

This hugely self-contained e-book approximately algebraic graph concept is written with the intention to retain the vigorous and unconventional surroundings of a spoken textual content to speak the passion the writer feels approximately this topic. the focal point is on homomorphisms and endomorphisms, matrices and eigenvalues. It ends with a tough bankruptcy at the topological query of embeddability of Cayley graphs on surfaces.

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**Example text**

6 Circulant graphs The so-called circulant graphs generalize, for example, cycles and complete graphs. Because of the circulant structure of their adjacency matrices, the computation of the characteristic polynomial is simpler than usual. Note, however, that the eigenvalues will not, in general, be real. 1. An n n matrix S is called a circulant matrix if its entries satisfy sij D s1j iC1 ; where the indices are reduced modulo n and thus belong to the set ¹1; : : : ; nº. In other words, row i of S can be obtained from row 1 of S via a circular shift of i 1 steps.

S1 ; : : : ; sn /. ; ! 2 ; : : : ; ! n 1 , where ! D exp 2n i , the nth roots of unity. They are pairwise distinct, so we get that W is diagonalizable. The eigenvalues of S are then determined by r D n X sj ! 6 Circulant graphs we get the eigenvalues r D n X aj ! j 1/r r D 0; : : : ; n ; 1: j D1 P P P Thus 0 D jnD1 aj D jnD2 aj and r D jnD11 aj C1 ! jr for r ¤ 0; see [Biggs 1996], p. 16, and, for example, p. 594 of [Brieskorn 1985]. 3. G/ is a circulant matrix. 1. 4 ([Cvetkovi´c et al. 6, p. ).

3. (a) For every k there exist cospectral k-tuples of regular, connected graphs. (b) Almost all (cf. 1 tp where sp is the number of trees with p vertices which are not cospectral to any other tree with p vertices, and tp is the number of trees with p vertices. See On the eigenvalues of a graph by A. J. Schwenk and R. J. 2 (with a sketched proof), in [Beineke/Wilson 1978]. 6. Eigenvalues, diameter and regularity The following theorem reveals an interesting connection between eigenvalues and the combinatorial structure of the graph.