Signature of Connected Line Graphs
Prior state unknown→disproved
no constant-bound repair of the conjecture is possible
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combinatorics / Spectral graph theory
Signature of Connected Line Graphs
Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A $14$-vertex witness has signature $2$, and chaining copies gives connected line graphs of signature $k + 1$ for every $k \ge 1$ - the signature is unbounded.
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Is the difference between the numbers of positive and negative adjacency eigenvalues of every connected line graph at most one? A $14$-vertex witness has signature $2$, and chaining copies gives connected line graphs of signature $k + 1$ for every $k \ge 1$ - the signature is unbounded.
no constant-bound repair of the conjecture is possible
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