Then g′ 3(x) = 5x4 −(4n+ 64)x 3 + (51n+ 225)x 2 −(196n+ 84)x+ 214n−348, g′′ 3(x) = 20x3 −(12n+ 192)x 2 + (102n+ 450)x−196n−84, g(3) 3 (x) = 60x2 −(24n+ 384)x+ 102n+ 450

Proof ofq(G 3)< q(K + 1,1,n−2): Note thatQ(G 3) has an equitable quotient matrixB 3 = n−2 2 2n−6 0 1 5 2 0 0 1 2 4 0 1 1 0 0 7 0 0 0 2 0 2 ! with CP g3(x) =x 5 −(n+ 16)x 4 + (17n+

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Signless Laplacian index conditions for trebly chorded cycles in graphs with given order

math.CO · 2026-04-17 · unverdicted · novelty 4.0

Graphs on n vertices with signless Laplacian index at or above a threshold depending on n contain a trebly chorded cycle unless they are two exceptional graphs.

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Signless Laplacian index conditions for trebly chorded cycles in graphs with given order math.CO · 2026-04-17 · unverdicted · none · ref 24
Graphs on n vertices with signless Laplacian index at or above a threshold depending on n contain a trebly chorded cycle unless they are two exceptional graphs.

Then g′ 3(x) = 5x4 −(4n+ 64)x 3 + (51n+ 225)x 2 −(196n+ 84)x+ 214n−348, g′′ 3(x) = 20x3 −(12n+ 192)x 2 + (102n+ 450)x−196n−84, g(3) 3 (x) = 60x2 −(24n+ 384)x+ 102n+ 450

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