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Given an adjacency matrix of a graph $G$, I was asked to do the following without drawing the graph:

A) Find the vertex of largest degree.

B) Does the graph have an Euler Circuit?

C) Is the graph Planar?

I did A and B. How can one justify if the graph is planar from adjacency matrix?

nonuser
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User8976
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2 Answers2

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There are some simple conditions which, if transgressed, would mean that the graph is non-planar but in general there is nothing which could easily determine whether the graph is planar.

Since you said 'without drawing the graph' can we assume that the matrix is fairly small?

For a small number of vertices (e.g. $6$) it is easy to check whether or not the graph contains $K_{3,3}$ or a subgraph homeomorphic to $K_5$.

The reference you have already been given

https://en.wikipedia.org/wiki/Planar_graph

should be useful for the simple conditions.

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If you have adjancency matrix like this one $$A = \pmatrix{0&1&1&1&1&1 &1\\1&0&1&1&1&1&1\\1&1&0&1&1&1&1 \\1&1&1&0&1&1&1 \\1&1&1&1&0&1&1\\1&1&1&1&1&0&1\\1&1&1&1&1&1&0}$$ then this graph is clearly not planar since all degrees are $6$ and in planar graph must exists vertex of degree less than $6$.

Probably you have some matrix from which is easy to deduce some numercial estimation which says that it is not planar, like $e> 3n-6$ or something like this.

nonuser
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