Narrow Results

Center function identifies vertices on a graph which minimizes the maximum distance to a set of input vertices. Axiomatic approach attempts to uniquely distinguish location functions such as center, median and mean functions using a set of axioms. In this paper, an attempt is made to find an axiomatic characterization of center function on crown graphs using both universal and nonuniversal axioms.

A mean of a sequence π = (x₁, x₂, …, x_k) of elements of a finite metric space (X, d) is an element x for which is minimum. The function Mean whose domain is the set of all finite sequences on X and is defined by Mean (π) = {x|x is a mean of π} is called the mean function on X. In this note, the mean function on finite trees is characterized axiomatically.

A median of a sequence π = (x₁, x₂, …, x_k) of elements of a finite metric space (X, d) is an element x for which is minimum. The function with domain the set of all finite sequences on X and defined by Med(π) = {x | x is a median of π} is called the Median function on X. In this note, an axiomatic characterization of the median function on finite trees is given.

A median of a sequence π = ( x₁, x₂, …, x_k) of elements of a finite metric space (X, d) is an element x ∈ X for which is a minimum. The function Median with domain the set of all finite sequences on X and defined by Med(π) = {x : x is a median of π} is called the median function on X. In this paper, the median function on finite Boolean lattices is axiomatically characterized via location functions.

An antimean of a sequence π = ( x₁, x₂, …, x_k) of elements of a finite metric space (X, d) is an element x for which is maximum. The function antimean with domain the set of all finite sequences on X and defined by AMean(π) = {x : x is an antimean of π} is called the antimean function on X. In this paper, the antimean function on finite paths is characterized.

In previous work, two axiomatic characterizations were given for the median function on median graphs: one involving the three simple and natural axioms anonymity, betweenness and consistency; the other involving faithfulness, consistency and -Condorcet. To date, the independence of these axioms has not been a serious point of study. The aim of this paper is to provide the missing answers. The independent subsets of these five axioms are determined precisely and examples provided in each case on arbitrary median graphs. There are three cases that stand out. Here nontrivial examples and proofs are needed to give a full answer. Extensive use of the structure of median graphs is used throughout.

Let $p$ be an integer such that $p\ge 1$. A $p$-value of a sequence $\pi =(x_1,x_2,\dots ,x_k)$ of elements of a finite metric space $(X,d)$ is an element $x\in X$ for which ${\sum }_{i=1}^k{d}^p(x,x_i)$ is minimum. The ${\ell }_p$ function whose domain is the set of all finite sequences on $X$, and defined by ${\ell }_p(\pi )=\{x:x$ is a $p$-value of $\pi \}$ is called the ${\ell }_p$ function on $X$. In this note, an axiomatic characterization of the ${\ell }_p$ function on finite Boolean lattices is presented.