13 papers · 1 filter
Proving a conjecture concerning chromatic number, size and least eigenvalue
Leyou Xu, Bo Zhou
Let be a simple nonempty graph with size , chromatic number , and least eigenvalue . We prove that \[ χ(χ-1) \le (m+1-λ^2)+\sqrt{(m+1-λ^2)^2-4(λ^2-1)(λ^2-m)} \] with e…
The number of cycles of a given length in dense hamiltonian graphs: proving Hilton's conjecture
Chengli Li, Leyou Xu, Bo Zhou
A classical theorem of Sheehan in 1977 states that every hamiltonian graph of order satisfying contains at least two cycles…
On -connected vertex-pancyclic graphs without pancyclic edges
Leyou Xu, Bo Zhou
An edge of a graph of order is pancyclic if it lies in a cycle of every length . A graph of order is vertex-pancyclic if every vertex lies in a cycle of every l…
Extremal -index problem in outerplanar graphs
Jin Cai, Leyou Xu, Bo Zhou
Outerplanar Turán problem has received considerable attention recently. We study the spectral version via -index. We determine the unique graph that maximizes the -index amon…
Girth and Laplacian eigenvalue distribution
Leyou Xu, Bo Zhou
Let be a connected graph of order with girth . For , let be the number of Laplacian eigenvalues (counting multiplicities) of tha…
Cycles and paths through specified vertices in graphs with a given clique number
Chengli Li, Leyou Xu
B. Bollobás and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least in any -connected graph of order…