Problems

Conjecture. If G ∈ G(k,n) and if p= lfloorn/2⌋ orlceiln/2⌉ ifn ≡ 1(bmod 4),k iseven;n ≡ 3(bmod 4),k iseven, lfloorn/2⌋ ifn ≡ 1(bmod 4),k isodd, n-2/2orn+2/2 ifn ≡ 2(bmod 4),k iseven, lceiln/2⌉ ifn ≡ 3(bmod 4),k isodd, then R^′(G)≥…

To replace this conjecture, a new conjecture is formulated that we coin as the Brualdi-Li tally conjecture.

Conjecture 1. Given an undirected connected graph G. We consider v an extremum of the Fiedler vector of the graph G.\tilde{G} is the graph obtained from G and v as in Proposition 1. Then for all x>0 the Fiedler vector \Phi(x,\cdot) of…

There exists a chain complex of finitely generated FI-modules C_* such that H_k(C_*) = H_k(K_p(S_•)).

We conjecture that this is true for all q.

In which condition power graph P(G) of a non-degenerate gyrogroup G is complete?

Let t_k,q(n) be defined by (5.1) t_k,q(j) = q^j, for 0 ≤ j ≤ k-1 t_k,q(n) = q ∑_l=0^k-2 (q-1)^l t_k,q(n-(l+2)), for n ≥ k. Then, S_F_q(T_2,3,…,k(n)) = t_k,q(n) for all values of n ≥ k.

Let C be a projective [n,k]_q three-weight code with non-zero weights w_1 < w_2 < w_3 satisfying w_1 + w_2 + w_3 = 3(1-1/q)n. Then w_2 = (1-1/q)n. Moreover, w_1 = w_2 - t and w_3 = w_2 + t, where t is a power of the characteristic p of F_q.

Based upon the results generated from our Sage script, we submit as a conjecture that these graphs constructed be the smallest graphs (by order) that have characteristic-dependent well-covered dimension for any given characteristic.

Note that the divisibility conditions in (22) should be equivalent to those in (23) if a t-(n,k,λ) exists. It is open if they are equivalent.