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I was going through the concept of graphs as matroids and I came upon the rank of a graph. Wikipedia lists it as $n - c$, $n = |V|$, $c =$ # of connected components.

I do understand rank and nullity of matrices, and graphs when expressed in their incidence matrix form have a one-to-one correspondence with the rank of its incidence matrix. However, I am not understanding how $r(G) = |V| - c$, $c = $ # of connected components and the definition of rank as the maximum size of a subforest of $G$ are equivalent.

I tried looking it up online but found no satisfactory explanation. Any resources that would be helpful to understand the concept would be great.

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  • $\begingroup$ Are you familiar with the definitions of (i) rank of a matroid, (ii) graphical matroid? $\endgroup$
    – Yuval Filmus
    Apr 10, 2022 at 5:41
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    $\begingroup$ The rank of a graphical matroid is the maximum number of edges which do not close a cycle. In a connected component of size $m$, the maximum number of edges which do not close a cycle is $m - 1$. Summing this over all connected components gives $n - c$. $\endgroup$
    – Yuval Filmus
    Apr 10, 2022 at 5:42
  • $\begingroup$ Thanks for this. This view of summing over all connected components makes everything clear. $\endgroup$
    – Ash Ketchum
    Apr 10, 2022 at 8:17

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The rank of a graphical matroid is the size of a spanning forest, which consists of a spanning tree in each connected component. A spanning tree for a connected component of size $m$ contains $m - 1$ edges. Summing this over all connected components, we see that a spanning forest contains $n - c$ edges.

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