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The number of trees in a graph | | Dhruv Mubayi
; Jacques Verstraete
; | | Date: |
23 Nov 2015 | | Abstract: | Let $T$ be a tree with $t$ edges. We show that the number of isomorphic
(labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$
is at least [2|E| prod_{v in V} (d(v) - t + 1)^{frac{(t-1)d(v)}{2|E|}}.]
Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at
least $t$ contains at least
$$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$.
This answers a question of Dellamonica et. al. (where the above statement was
proved when $T$ is the path with three edges) while extending an old result of
ErdH os and Simonovits. | | Source: | arXiv, 1511.7274 | | Services: | Forum | Review | PDF | Favorites | |
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