The energy of a graph $G$ is the sum absolute values eigenvalues adjacency matrix $G$. Let $s^+(G), s^-(G)$ denote squares positive and negative $G$, respectively. It was conjectured by [Elphick, Farber, Goldberg, Wocjan, Discrete Math. (2016)] that if connected order $n$, then $s^+(G)\geq n-1$ $s^-(G) \geq n-1$. In this paper, we show partial results towards conjecture. particular, numerous st...

A conjecture of Alon, Pach and Solymosi, which is equivalent to the celebrated Erdős–Hajnal Conjecture, states that for every tournament S there exists ɛ(S)>0 such if T an n-vertex does not contain as a subtournament, then contains transitive subtournament on at least nɛ(S) vertices. Let C5 be unique five-vertex where vertex has two inneighbors outneighbors. The Alon–Pach–Solymosi known true ca...

They also conjectured that for Jki<feâ£o, pk!(n)Spk2(n) and for ko<h<k2i pki(n)<pk2(n). They verify this conjecture for n^32. Recently Todd published a table of all the pkin) for n^lOO, and it is easy to verify the conjecture for w^glOO. I am unable to prove or disprove this conjecture. They also remark that pk0{n) differs from c-inm i0g n by i e s s than 1 for n^ 32; (2) shows that for large n...

0. Introduction Let X be a variety of logarithmic general type, deened over a number eld K. Let S be a nite set of places in K and let O K;S be the ring of S-integers. Suppose that X is a model of X over Spec O K;S. As a natural generalizasion of theorems of Siegel and Faltings, It was conjectured by S. Lang and P. Vojta ((Vojta], conjecture 4.4) that the set of S-integral points X(O K;S) is no...

We prove a dual version of the celebrated inequality of D. Reimer (a.k.a. the van den Berg-Kesten conjecture). We use the dual inequality to prove a combinatorial conjecture of S. Rudich motivated by questions in cryptographic complexity. One consequence of Rudich’s Conjecture is that there is an oracle relative to which one-way functions exist but one-way permutations do not. The dual inequali...

He proved in [3] this conjecture in the case of the second and third punctual Hilbert scheme of an algebraic K3 surface. In this paper, we observe that the results of [5] can lead to a more general conjecture concerning the Chow ring of an algebraic hyper-Kähler variety. Namely, the full statement of Theorem 0.1 can be interpreted by saying that any polynomial relation between [c2(TS)], [c1(Li)...

A famous conjecture of Gyárfás and Sumner states for any tree T and integer k, if the chromatic number of a graph is large enough, either the graph contains a clique of size k or it contains T as an induced subgraph. We discuss some results and open problems about extensions of this conjecture to oriented graphs. We conjecture that for every oriented star S and integer k, if the chromatic numbe...