4 research outputs found
On unbalanced Boolean functions with best correlation immunity
It is known that the order of correlation immunity of a nonconstant
unbalanced Boolean function in variables cannot exceed ; moreover,
it is if and only if the function corresponds to an equitable
-partition of the -cube with an eigenvalue of the quotient matrix.
The known series of such functions have proportion , , or of
the number of ones and zeros. We prove that if a nonconstant unbalanced Boolean
function attains the correlation-immunity bound and has ratio of the
number of ones and zeros, then is divisible by . In particular, this
proves the nonexistence of equitable partitions for an infinite series of
putative quotient matrices. We also establish that there are exactly
equivalence classes of the equitable partitions of the -cube with quotient
matrix and classes, with . These
parameters correspond to the Boolean functions in variables with
correlation immunity and proportion and , respectively (the case
remains unsolved). This also implies the characterization of the
orthogonal arrays OA and OA.Comment: v3: final; title changed; revised; OA(512,11,2,6) discusse
Some results on the Wiener index related to the \v{S}olt\'{e}s problem of graphs
The Wiener index, , of a connected graph is the sum of distances
between its vertices. In 2021, Akhmejanova et al. posed the problem of finding
graphs with large . It is shown that there is a graph with for any
integer . In particular, there is a regular graph of even degree with
this property for any odd . The proposed approach allows to construct
new families of graphs with when the order of
increases.Comment: 9 pages, 3 tables, 7 figure