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I have just started studying graph theory and having trouble with understanding the difference b/w isomorphism and equality of two graphs.According what I have studied so far, I am able to conclude that isomorphic graphs can have same diagrams when represented on paper, but equal graphs also have same diagram on paper, if that is so, then what is the difference b/w equality and isomorphism.

It would be really helpful, if a person could throw some light on what are the difference b/w the above two terms. Specifically, any example of an isomorphic graph which is not equal.

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By definition a graph is a set of edges $E\subseteq V^2$ and vertices. An other graph

$\bar E\subseteq \bar V^2$ is equal if $E=\bar E$ and $V=\bar V$, but isomorphic if there exists a bijection $f:V\rightarrow \bar V$ such that

$(x,y)\in E \Rightarrow (f(x),f(y))\in \bar E$.

Isomorphic is as close as can be when the graphs not have identical sets of edges and vertices.

This two graphs are isomorphic, but not equal. In applications where the graphs are LABELLED, the useful concept might be equality and a mere isomorphism may not be enough.

Mom Dad

```
\ /
Son
```

Son Dad

```
\ /
Mom
```

A graph is a set of vertices and edges. Isomorphic graphs *look* the same but aren’t. For example, the persons in a household can be turned into a graph by decalring that there is an edge $ab$ whenever $a$ is parent or child of $b$. So the graph of the inhabitants of a certin house in Evergreen terrace, Springfield, consists of five vertices where each of “Marge” and “Homer” is directly connected with each of “Bart”, “Lisa”, “Maggie”. We would obtain an *isomorphic* graph with any other typical couple-with-three-kids household, but not the *same* (i.e. not all these graphs will contain a vertex named Homer).

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