I don't see any real reason why we would use an AM to represent a graph, beside visual appeal and ease. Generally, we would perform matrix operations on Matrices like |A|, Transpose and loads of other things but that magic doesn't seem to add up to what I am looking at, AM. Anyone care to explain or give me some intuition so I may respect AM a bit more and use it to study the graph at hand purely by playing around with the numbers. Obviously, you get the point.
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1See e.g. matrix tree theorem – Grigory M Jan 19 '14 at 11:11
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Thanks, looks interesting. – Ali Gajani Jan 19 '14 at 11:14
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There's an entire branch of mathematics devoted to the study of matrix representations of graphs. Applications are abound. A rather stunning application of adjacency matrices occur in understanding which parts of a network are the most central. This links directly to things like Google's PageRank algorithm for example. This question also seems to be related. – EuYu Jan 19 '14 at 11:28
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Thanks for your answer EuYu. I read somewhere you could find some interesting things by doing A transpose, not sure though. – Ali Gajani Jan 19 '14 at 11:39
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1See http://math.stackexchange.com/questions/308952/motivation-for-spectral-graph-theory/312555#312555 for relevant discussion. – Chris Godsil Jan 20 '14 at 13:47
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Here is one use of adjacency matrices. Let $G=(V,E)$ be a graph and let $A(G)=\{a_{ij}\}$ be its adjacency matrix. Then the $ij$-th entry of $A(G)^n$ is the number of walks of length $n$ starting at the $i$-th vertex that end at the $j$-th vertex.
For large $n$, it's inefficient to calculate the $n$-th power of $A(G)$, so we find its eigenvalues to diagonalize $A(G)$: $$A(G)=P^TDP$$ so that $A(G)^n=P^TD^n P$ can be easily calculated.
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