Using the AutoGraphiX 2 system, Aouchiche, Hansen and Zheng [2] proposed a conjecture that the difference and the ratio of the Randić index and the diameter of a graph are minimum for paths. We prove this conjecture for chemical graphs.

It is a longstanding open problem to devise an oracle relative to which BQP does not lie in the Polynomial-Time Hierarchy (PH). We advance a natural conjecture about the capacity of the NisanWigderson pseudorandom generator [NW94] to fool AC0, with MAJORITY as its hard function. Our conjecture is essentially that the loss due to the hybrid argument (which is a component of the standard proof fr...

In a paper of Kedlaya and Medvedovsky [KM19] , the number distinct dihedral mod 2 modular representations prime level N was calculated, conjecture on dimension space weight forms giving rise to each representation stated. this we prove conjecture.

A refinement of the rank 1 abelian Stark conjecture has been formulated by B. Gross. This conjecture predicts some p-adic analytic nature of a modification of the Stark unit. The conjecture makes perfect sense even when p is an archimedean place. Here we consider the conjecture when p is a real place, and interpret it in terms of 2-adic properties of special values of L-functions. We prove the ...

Abstract: The well known Zarankiewicz’ conjecture is said that the crossing number of the complete bipartite graph Km,n (m ≤ n) is Z(m, n), where Z(m,n) = ⌊ m 2 ⌋⌊ 2 ⌋⌊ 2 ⌋ ⌊ 2 ⌋ (for any real number x, ⌊x⌋ denotes the maximal integer no more than x). Presently, Zarankiewicz’ conjecture is proved true only for the case m ≤ 6. In this article, the authors prove that if Zarankiewicz’ conjecture h...

We consider the conjecture stating that a matrix with rank o(n) and ones on the main diagonal must contain nonzero entries on a 2 2 submatrix with one entry on the main diagonal. We show that a slightly stronger conjecture implies that an explicit linear transformation cannot be computed by linear size and logarithmic depth circuits. We prove some partial results supporting the conjecture.

A famous conjecture (usually called Ryser’s conjecture), appeared in the Ph.D thesis of his student, J. R. Henderson [9], states that for an r-uniform r-partite hypergraph H, the inequality τ(H) ≤ (r − 1)·ν(H) always holds. This conjecture is widely open, except in the case of r = 2, when it is equivalent to Kőnig’s theorem [16], and in the case of r = 3, which was proved by Aharoni in 2001 [2]...