Does the leading eigenvalue always increase with an edge addition to the graph? If so, how can I prove this?
Thank you
2026-03-26 22:53:27.1774565607
Does the leading eigenvalue of a connected undirected graph always increase with an edge addition?
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It need not increase. Consider for example the graph $G$ formed by taking the disjoint union of a $k$-regular graph and two isolated vertices $x,y$.
The maximal eigenvalue of $G$ and $G+xy$ is clearly $k$.
If you're looking for ways to tackle this problem I suppose you should look at Weyl's inequalities.