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# 数学代写|图论代考GRAPH THEORY代写|Isomorphism

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## 数学代写|图论代写GRAPH THEORY代考|Isomorphism

Two graphs $G_1$ and $G_2$ are isomorphic if there is a one-one correspondence between the vertices of $G_1$ and those of $G_2$ such that the number of edges joining any two vertices of $G_1$ equals the number of edges joining the corresponding vertices of $G_2$. For example, the two graphs in Fig. 1.3 are isomorphic, under the correspondence
$$u \leftrightarrow I, v \leftrightarrow m, w \leftrightarrow n, x \leftrightarrow p, y \leftrightarrow q, z \leftrightarrow r .$$

For many problems, the labels on the vertices are unnecessary and we drop them. We then say that two ‘unlabelled graphs’ are isomorphic if we can assign labels to their vertices so that the resulting ‘labelled graphs’ are isomorphic. For example, we regard the unlabelled graphs in Fig. 1.4 as isomorphic, since the labelled graphs in Fig. 1.3 are isomorphic.

The difference between labelled and unlabelled graphs becomes more apparent when we try to count them. For example, if we restrict ourselves to graphs with three vertices, then there are eight different labelled graphs (see Fig. 1.5), but only four unlabelled ones (see Fig. 1.6). It is usually clear from the context whether we are referring to labelled or unlabelled graphs.

## 数学代写|图论代写GRAPH THEORY代考|Connected graphs

We can combine two graphs to make a larger graph. If the two graphs are $G_1$ and $G_2$ and their vertex-sets $V\left(G_1\right)$ and $V\left(G_2\right)$ are disjoint, then their union $G_1 \cup G_2$ is the graph with vertex-set $V\left(G_1\right) \cup V\left(G_2\right)$ and edge-family $E\left(G_1\right) \cup E\left(G_2\right)$ (see Fig. 1.7).
Most of the graphs discussed so far have been ‘in one piece’. A graph is connected if it cannot be expressed as a union of graphs, and disconnected otherwise. Clearly, any disconnected graph $G$ can be expressed as the union of connected graphs, each of which is called a component of $G$, a disconnected graph with three components is shown in Fig. 1.8.

When proving results about graphs in general, we can often obtain the corresponding results for connected graphs and then apply them to each component separately. A table of all the unlabelled connected simple graphs with up to five vertices is given in Fig. 1.9.

We say that two vertices $v$ and $w$ of a graph are adjacent if there is an edge $v w$ joining them, and the vertices $v$ and $w$ are then incident with such an edge. We also say that two distinct edges $e$ and $f$ are adjacent if they have a vertex in common (see Fig. 1.10).

The degree of a vertex $v$ is the number of edges incident with $v$, and is written $\operatorname{deg}(v)$; when calculating the degree of $v$, we usually make the convention that a loop at $v$ contributes 2 (rather than 1 ) to $\operatorname{deg}(v)$. A vertex of degree 0 is an isolated vertex and a vertex of degree 1 is an end-vertex. Thus each of the two graphs in Fig. 1.11 has two end-vertices and three vertices of degree 2, while the graph in Fig. 1.12 has one end-vertex, one vertex of degree 3 , one of degree 6 and one of degree 8 .

The degree sequence of a graph consists of the degrees written in increasing order, with repeats where necessary. For example, the degree sequences of the graphs in Figs 1.11 and 1.12 are $(1,1,2,2,2)$ and $(1,3,6,8)$.

## 数学代写|图论代写GRAPH THEORY代考|Isomorphism

$$u \leftrightarrow I, v \leftrightarrow m, w \leftrightarrow n, x \leftrightarrow p, y \leftrightarrow q, z \leftrightarrow r .$$

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