# On Central-Peripheral Appendage Numbers of Uniform Central Graphs

## TL;DR

This paper introduces the central-peripheral appendage number for uniform central graphs, providing formulas based on graph parameters and establishing bounds, including a maximum of 6 when the diameter exceeds 2.

## Contribution

It defines and computes the central-peripheral appendage number for UCGs, offering new formulas and structure theorems based on graph properties.

## Key findings

- A_{ucg}(C, P) is computed using radius and diameter of P and the completeness of C.
- A_{ucg}(C, P) ; 6 when diam(P) > 2.
- Structural characterizations of UCGs are provided.

## Abstract

In a uniform central graph (UCG) the eccentric verticies of a central vertex is the same for all central verticies. This collection of eccentric verticies is the centered periphery. For a pair of graphs $(C, P)$ the central-peripheral appendage number, $A_{ucg}(C, P)$, is the minimum number verticies needed to be adjoined to the graphs $C$ and $P$ in order to construct a uniform central graph H with center C and centered-periphery P. We compute $A_{ucg}(C, P)$ in terms of the radius and diameter of P and whether or not $C$ is a complete graph. In the process we show $A_{ucg}(C, P)\leq 6$ if $diam(P) > 2$. We also provide structure theorems for UCGs in terms of the centered periphery.