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Generalities on Graphs

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Topological Crystallography

Part of the book series: Surveys and Tutorials in the Applied Mathematical Sciences ((STAMS,volume 6))

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Abstract

A graph is an abstract concept describing a set of objects where some pairs of objects are connected by links.As mentioned in Notes in the previous chapter, the notion of graphs was implicitly used by Leonhard Euler (1707–1783). Practically, we represent a graph by a diagram in a plane or space consisting of points (vertices) and lines (edges). Lines are allowed to intersect each other. In some applications, say communication networks and electric circuits, a diagram may be associated in a direct way with the objects considered. A crystal structure also provides us with a visible diagram in space when we represent atoms and chemical bonds by points and lines,Strictly speaking, the line representing a bond is virtual. See the beginning part of Chap. 7. respectively. In many other cases, however, a graph structure is not explicitly visible at first hand.

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Notes

  1. 1.

    As mentioned in Notes in the previous chapter, the notion of graphs was implicitly used by Leonhard Euler (1707–1783).

  2. 2.

    Strictly speaking, the line representing a bond is virtual. See the beginning part of Chap. 7.

  3. 3.

    Uber die Grundlagen der Geometrie, 1899.

  4. 4.

    In Euclid’s Elements, we find the following definitions. “A point is that which has no part”. “A line is breadthless length”. “A straight line is a line which lies evenly with the point on itself”.

  5. 5.

    I am, of course, not denying the significance of intuition in ongoing studies. Without intuitive thinking, theorems in geometry could not have been formulated. Also we must emphasize the importance of pictures in understanding any “formal” proof.

  6. 6.

    When a molecule is formed by atoms, the electron cloud of one atom and the electron cloud of another atom interact to form a new electron cloud. This formation, depending on the orbitals overlapping, is known as hybridisation, and is classified as sp, sp 2 and sp 3, etc.

  7. 7.

    In the case that X is an infinite graph, it is often natural to think of Aut(X) as a topological group with “compact-open topology” (see Sect. 9.6).

  8. 8.

    For instance, the honeycomb lattice depicted in Fig. 3.7 appears as the crystal structure of graphene, an allotrope of carbon. The term graphene was coined by Hanns-Peter Boehm (1962). The Nobel Prize in Physics for 2010 was awarded to Andre Geim and Konstantin Novoselov “for groundbreaking experiments regarding the two-dimensional material graphene”.

  9. 9.

    This is also called a tortoise shell pattern.

  10. 10.

    Such a lattice is called a period lattice of the honeycomb lattice.

  11. 11.

    Graphite as a mineral is one of the allotropes of carbon, and the most stable form under standard conditions. It has a layered, planar structure. In each layer, the carbon atoms are arranged in a honeycomb lattice.

  12. 12.

    When X is infinite, we need Zorn’s lemma to deduce the existence. Zorn’s lemma is known to be equivalent to Axiom of Choice.

  13. 13.

    The even or odd quality of an integer.

  14. 14.

    See Example 6.5 in Sect. 6.2 and Fig. 8.1 in Sect. 8.3.

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Sunada, T. (2013). Generalities on Graphs. In: Topological Crystallography. Surveys and Tutorials in the Applied Mathematical Sciences, vol 6. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54177-6_3

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