The Distance Matching Extension in K_(1,k)-free Graphs with High Local Connectedness

A matching is extendable in a graph G if G has a perfect matching containing it.A distance q matching is a matching such that the distance between any two distinct matching edges is at least q.In this paper,we prove that any distance 2k-3 matching is extendable in a connected and locally(k-1)-connected K1,k-free graph of even order.Furthermore,we also prove that any distance q matching M in an r-connected and locally(k-1)-connected K1,k-free graph of even order is extendable provided that|M|is bounded by a function on r,k and q.Our results improve some results in[J.Graph Theory 93(2020),5–20].

REMARKS ON JOHN DISKS

Let D■R2 be a Jordan domain,D*=R2\D,the exterior of D.In this article,the authors obtained the following results:(1)If D is a John disk,then D is an outer linearly locally connected domain;(2)If D* is a John disk,then D is an inner linearly locally connected domain;(3)A homeomorphism f:R 2 →R 2 is a quasiconformal mapping if and only if f(D)is a John disk for any John disk D■R 2 ;and(4)If D is a bounded quasidisk,then D is a John disk,and there exists an unbounded quasidisk which is not a John disk.

Connectedness of Invariant Sets of Graph-Directed IFS

In this paper, we study the connectedness of the invariant sets of a graph-directed iterated function system（IFS）. For a graph-directed IFS with N states, we construct N graphs. We prove that all the invariant sets are connected, if and only if all the N graphs are connected; in this case, the invariant sets are all locally connected and path connected. Our result extends the results on the connectedness of the self-similar sets.

Two Kinds Equal Frequency Circuits to Achieve Locally Resonant Band Gap of a Circular Plate Attached Alternately by Piezoelectric Unimorphs

A circular thin plate is proposed for vibration attenuation,which is attached alternately by annular piezoelectric unimorphs with resonant shunt circuits.Two kinds of equal frequency resonant shunt circuits are designed to achieve an integrated locally resonant（LR）band gap（BG） with a much smaller transmission factor：（1） the structure is arrayed periodically while the resonant shunt circuits are aperiodic;（2） the resonant shunt circuits are periodic while the structure is aperiodic.The transmission factor curve is calculated,which is validated by the finite element method.Dependences of the LR BG performance upon the geometric and electric parameters are also analyzed.

Claw─centre Independent Graphs

gives that every connected,locally connected K 1,4 free,almost claw free graph on at least three vertices is vertices pancyclic. This paper shows that every connected,locally 2 connected K 1,4 free claw centre independent graph on at least three vertices is vertex pancyclic.