The Simultaneous Fractional Dimension of Graph Families

作     者：Cong X.KANG Iztok PETERIN Eunjeong YI Cong X.KANG;Iztok PETERIN;Eunjeong YI

作者机构：Texas A&M University at GalvestonGalvestonTX 77553USA University of MariborFEECS Smetanova 172000 MariborSlovenia 

出 版 物：《Acta Mathematica Sinica,English Series》 (数学学报（英文版）)

年 卷 期：2023年第39卷第8期

页      面：1425-1441页

核心收录：

学科分类：07[理学] 070104[理学-应用数学] 0701[理学-数学] 

基　　金：Supported by US-Slovenia Bilateral Collaboration Grant(BI-US/19-21-077) 

主　　题：Metric dimension fractional metric dimension resolving function simultaneous(metric)dimension simultaneous fractional(metric)dimension 

摘      要：For a connected graph G with vertex set V,let RG{x,y}={z∈V:dG(x,z)≠dG(y,z)}for any distinct x,y∈V,where dG(u,w)denotes the length of a shortest uw-path in *** a real-valued function g defined on V,let g(V)=∑s∈V g(s).Let C={G_(1),G_(2),...,G_(k)}be a family of connected graphs having a common vertex set V,where k≥2 and|V|≥3.A real-valued function h:V→[0,1]is a simultaneous resolving function of C if h(RG{x,y})≥1 for any distinct vertices x,y∈V and for every graph G∈*** simultaneous fractional dimension,Sdf(C),of C is min{h(V):h is a simultaneous resolving function of C}.In this paper,we initiate the study of the simultaneous fractional dimension of a graph *** obtain max1≤i≤k{dimf(Gi)}≤Sd_(f)(C)≤min{∑k i=1 dimf(Gi),|V|/2},where both bounds are *** characterize C satisfying Sdf(C)=1,examine C satisfying Sdf(C)=|V|/2,and determine Sdf(C)when C is a family of vertex-transitive *** also obtain some results on the simultaneous fractional dimension of a graph and its complement.