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co.combinatorics – Relation between two conjectures on reconstruction of graphs
In spectral graph concept, there’s a surmise that claims: Almost each graph is clear by its adjacency spectrum ($DS$). This surmise belongs to professor Willem Haemers.
Also, we’ve got a outcome by professor Béla Bollobás that claims: Almost each graph has reconstruction quantity three ($RC=$ Almost each graph is reconstructible).
We have a theorem in spectral graph concept that claims: We can assemble the attribute polynomial of a graph by its deck. I’ve two questions:
$1)$ Can we are saying one thing love that: Almost each graph has a attribute polynomial reconstruction quantity $ok < n$? For instance $ok=3$.
By attribute polynomial reconstruction quantity $3$, I denote if we’ve got a $3$ apt graphs within the deck of the unique graph, we will secure its attribute polynomial.
$2)$ What is the issue if we are saying that $RCrightarrow DS$ or $DSrightarrow RC$? Is it as a result of we labor with the phrases “Almost every …”?
Thank you in forward.
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