Recursive Formulas for Beans Functions of Graphs

Abstract

In this paper, we regard each edge of a connected graph G as a line segment having a unit length, and focus on not only the "vertices" but also any "point" lying along such a line segment. So we can define the distance between two points on G as the length of a shortest curve joining them along G. The beans function B G (x) of a connected graph G is defined as the maximum number of points on G such that any pair of points have distance at least x > 0. We shall show a recursive formula for B G (x) which enables us to determine the value of B G (x) for all x ≤ 1 by evaluating it only for 1/2 < x ≤ 1. As applications of this recursive formula, we shall propose an algorithm for computing B G (x) for a given value of x ≤ 1, and determine the beans functions of the complete graphs K n .

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

“…Our goal is to introduce a mixed integer linear program (MILP) to compute the best polynomial for the bound (8) (and hence the same for the bound (18)). Since such a bound is also valid for the quantum k-independence number and this parameter is not computable in general, the use MILPs to find the best polynomial is justified.…”

“…The bound (18) can be strengthened when k = 1 and p k (A) = A as follows (see Elphick and Wocjan [17,Th. 1]).…”

“…Therefore, lower bounds on the k-distance chromatic number can be obtained by finding upper bounds on the k-independence number, and vice versa. The k-independence number has also been studied in several other contexts (see [6,16,24,25,19,43] for some examples) and it is related to other combinatorial parameters, such as the average distance [26], the strong chromatic index [41], the d-diameter [12], and to the beans function of a connected graph [18].…”

Optimization of eigenvalue bounds for the independence and chromatic number of graph powers

“…The parameter α k has also been studied in several other contexts (see [6,13,22,23,16,48] for some examples) and it is related to other combinatorial parameters, such as the average distance [24], the packing chromatic number [26], the injective chromatic number [29], the strong chromatic index [47] and the d-diameter [9]. Recently, the k-independence number has also been related to the beans function of a connected graph [15].…”