focalinterest.net

Tag: Discrete Mathematics & Combinatorics

  • Edge version of graph inducibility has polynomial growth

    What the study found

    The authors show that, for any graph H, the maximum number of induced copies in a graph with m edges grows like a constant times m raised to the fractional independence number of H. For cycles and paths, they give additional bounds and conjectures about the asymptotic constant.

    Why the authors say this matters

    The study suggests that understanding the edge-based version of inducibility can be reduced to a power-law growth rate determined by a graph invariant called the fractional independence number. The authors also indicate that the constant factor in front of this main term is an important remaining question for cycles and paths.

    What the researchers tested

    The researchers studied the edge version of inducibility, where the question is how many induced copies of a graph H can appear in a graph with exactly m edges. They used the entropy method and focused especially on cycles C_k and paths P_k.

    What worked and what didn't

    They prove that rho(H, m) = Theta(m^{alpha_f(H)}) for any graph H, where alpha_f(H) is the fractional independence number. For cycles, they conjecture an asymptotic formula for k >= 5, prove an upper bound with an extra constant factor for even cycles, and only an upper bound with an extra factor depending on k for odd cycles. For paths, they prove rho(P_{2l}, m) <= m^l / [2(l-1)^{l-1}] and rho(P_{2l+1}, m) <= m^{l+1} / [4l^l], and they also conjecture the asymptotic value of rho(P_k, m).

    What to keep in mind

    The abstract does not provide full proofs or specify whether the conjectures are resolved. The paper gives exact asymptotic order for all graphs, but the sharper constant-factor behavior is only partially established for cycles and paths.

    • For any graph H, the edge version of inducibility grows on the order of m raised to the fractional independence number of H.
    • The paper focuses on the constant factor in front of this growth rate for cycles and paths.
    • For cycles C_k with k >= 5, the authors conjecture an asymptotic formula and say the bound is achieved by the blow up of C_k.
    • For even cycles, the authors establish an upper bound with an extra constant factor; for odd cycles, the extra factor depends on k.
    • For paths P_{2l} and P_{2l+1}, the paper gives explicit upper bounds in terms of m and l.
    • The entropy method is the main tool used in the paper.

  • Subnormalisers of semisimple elements in finite groups are determined

    What the study found

    The study determines the subnormalisers of semisimple elements of prime power order in finite quasi-simple groups of Lie type. It also determines the maximal overgroups of normalisers of Sylow tori, where a Sylow torus is a maximal torus tied to a prime divisor condition in groups of Lie type.

    Why the authors say this matters

    The authors say the work is motivated by the recent character correspondence conjecture of Moretó and Rizo. They also state that it is motivated by the question of whether quasi-semiregular elements exist in finite permutation groups.

    What the researchers tested

    The paper studies finite quasi-simple groups of Lie type and examines semisimple elements of prime power order. The authors determine subnormalisers and investigate maximal overgroups of normalisers of Sylow tori.

    What worked and what didn't

    The abstract states that the subnormalisers are determined for the elements under study. It also states that the maximal overgroups of normalisers of Sylow tori are determined; it does not describe any unsuccessful cases or exceptions.

    What to keep in mind

    The available summary is limited to the abstract, so no detailed methods, examples, or exceptions are provided. The abstract does not state broader consequences beyond the motivating questions.

    • Subnormalisers of semisimple elements of prime power order are determined.
    • The setting is finite quasi-simple groups of Lie type.
    • Maximal overgroups of normalisers of Sylow tori are also determined.
    • The work is motivated by a character correspondence conjecture of Moretó and Rizo.
    • The abstract also mentions quasi-semiregular elements in finite permutation groups as motivation.

  • Exact residual finiteness growth for some two-step nilpotent groups

    What the study found

    The authors found an improved polylogarithmic upper bound for the residual finiteness growth of two-step nilpotent groups. They also show that this bound depends only on the complex Mal’cev completion of the group, and that it is exact when the commutator subgroup is one- or two-dimensional.

    Why the authors say this matters

    The authors note that exact asymptotics for residual finiteness growth are unknown for many groups, including general nilpotent groups. The findings suggest progress on that broader problem by giving a sharper bound in the two-step nilpotent case.

    What the researchers tested

    The researchers studied residual finiteness growth, a function that measures the size of a finite quotient needed to detect an element of bounded norm in a finitely generated residually finite group. They focused on two-step nilpotent groups and compared their results with bounds known in the literature.

    What worked and what didn't

    The improved polylogarithmic upper bound worked for all two-step nilpotent groups considered in the paper. The authors also proved exactness in the special cases where the commutator subgroup has dimension one or two. For the general nilpotent setting, the abstract says exact asymptotics are still unknown.

    What to keep in mind

    The abstract does not describe the proof details or the full range of assumptions beyond finitely generated residually finite groups. It also does not state whether the conjecture for the general case is proved, only that the authors conjecture their exactness result holds more broadly.

    • The paper gives an improved polylogarithmic upper bound for residual finiteness growth in two-step nilpotent groups.
    • The bound depends only on the complex Mal’cev completion of the group.
    • The bound is exact when the commutator subgroup is one- or two-dimensional.
    • The abstract says exact asymptotics remain unknown for many groups, including general nilpotent groups.
    • The authors conjecture that the exactness result extends beyond the special low-dimensional cases.