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On a theorem of Nosal
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abstract
Let $G$ be a graph with $m$ edges and spectral radius $\lambda_{1}$. Let $bk\left( G\right) $ stand for the maximal number of triangles with a common edge in $G$. In 1970 Nosal proved that if $\lambda_{1}^{2}>m,$ then $G$ contains a triangle. In this paper we show that the same premise implies that \[ bk\left( G\right) >\frac{1}{12}\sqrt[4]{m}. \] This result settles a conjecture of Zhai, Lin, and Shu. Write $\lambda_{2}$ for the second largest eigenvalue of $G$. Recently, Lin, Ning, and Wu showed that if $G$ is a triangle-free graph of order at least three, then \[ \lambda_{1}^{2}+\lambda_{2}^{2}\leq m, \] thereby settling the simplest case of a conjecture of Bollob\'{a}s and the author. We give a simpler proof of their result.
Forward citations
Cited by 5 Pith papers
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Supersaturation in Nosal graphs: Triangles and books
Every m-edge graph with spectral radius λ has at least m(λ−√m) triangles, and every Nosal graph contains a book of size > ¼√m and at least (1/8−o(1))m kites.
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Two problems on booksize and triangular edges in Nosal graphs
Any non-complete-bipartite graph with m edges and spectral radius at least √m has a book of size at least ρ(G)/3 and at least ρ(G) triangular edges.
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An edge-spectral supersaturation of Mubayi's theorem for color-critical graphs
For color-critical F with χ(F)=r+1≥4, λ²(G)≥2(1-1/r)m+q with 0<q≤δ_F√m forces at least (B_F-o(1))q m^{(f-2)/2} copies of F, and B_F is best possible.
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On a spectral booksize problem fo non bipartite graphs
Every sufficiently large m-edge non-bipartite graph without isolated vertices satisfying rho(G)^2 >= m-1+2/(rho(G)-1) is either an exceptional graph S+_{m,s} or contains a book of size at least (1/4-o(1)) sqrt(m), and...
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Spectral Tur\'{a}n problem of non-bipartite graphs: Forbidden books
For n≥8(r^2+r+4), the unique spectral extremal non-bipartite B_{r+1}-free graph of order n is the nearly balanced complete bipartite graph with one added vertex adjacent to r vertices in each part.
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