The well known Mantels Theorem states that a graph on $n$ vertices and $m$ edges contains a triangle if $m>frac{n^2}{4}$. Nosal proved that every graph on $m$ edges contains a triangle if the spectral radius $lambda_1>sqrt{m}$, which is a spectral an
alog of Mantels Theorem. Furthermore, by using Motzkin-Straus Inequality, Nikiforov sharped Nosals result and characterized the extremal graphs when the equality holds. Our first contribution in this note is to give two new proofs of the spectral concise Mantels Theorem due to Nikiforov (without help of Motzkin-Straus Inequality). Nikiforov also obtained some results concerning the existence of consecutive cycles and spectral radius. Second, we prove a theorem concerning the existence of consecutive even cycles and spectral radius, which slightly improves a result of Nikiforov. At last, we focus on spectral radius inequalities. Hong proved his famous bound for spectral radius. Later, Hong, Shu and Fang generalized Hongs bound to connected graphs with given minimum degree. By using quite different technique, Nikiforov proved Hong et al.s bound for general graphs independently. In this note, we prove a new spectral inequality by applying the technique of Nikiforov. Our result extends Stanleys spectral inequality.
Recently, Straub gave an interesting $q$-analogue of a binomial congruence of Ljunggren. In this note we give an inductive proof of his result.
A graph $G$ on $n$ vertices is Hamiltonian if it contains a spanning cycle, and pancyclic if it contains cycles of all lengths from 3 to $n$. In 1984, Fan presented a degree condition involving every pair of vertices at distance two for a 2-connected
graph to be Hamiltonian. Motivated by Fans result, we say that an induced subgraph $H$ of $G$ is $f_1$-heavy if for every pair of vertices $u,vin V(H)$, $d_{H}(u,v)=2$ implies $max{d(u),d(v)}geq (n+1)/2$. For a given graph $R$, $G$ is called $R$-$f_1$-heavy if every induced subgraph of $G$ isomorphic to $R$ is $f_1$-heavy. In this paper we show that for a connected graph $S$ with $S eq P_3$ and a 2-connected claw-$f_1$-heavy graph $G$ which is not a cycle, $G$ being $S$-$f_1$-heavy implies $G$ is pancyclic if $S=P_4,Z_1$ or $Z_2$, where claw is $K_{1,3}$ and $Z_i$ is the path $a_1a_2a_3... a_{i+2}a_{i+3}$ plus the edge $a_1a_3$. Our result partially improves a previous theorem due to Bedrossian on pancyclicity of 2-connected graphs.
We consider the refinement of the holographic entanglement entropy for the holographic dual theories to the AdS solitons and AdS black holes, including the corrected ones by the Gauss-Bonnet term. The refinement is obtained by extracting the UV-indep
endent piece of the holographic entanglement entropy, the so-called renormalized entanglement entropy which is independent of the choices of UV cutoff. Our main results are (i) the renormalized entanglement entropies of the AdS$_{d+1}$ soliton for $d=4,5$ are neither monotonically decreasing along the RG flow nor positive definite, especially around the deconfinement/confinement phase transition; (ii) there is no topological entanglement entropy for AdS$_5$ soliton even with Gauss-Bonnet correction; (iii) for the AdS black holes, the renormalized entanglement entropy obeys an expected volume law at IR regime, and the transition between UV and IR regimes is a smooth crossover even with Gauss-Bonnet correction; (iv) based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state.
