Vector Valued Switching in Signed Graphs

TL;DR

This paper introduces vector valued switching functions in signed graphs, defining new dimensions to classify imbalance levels, and provides bounds and calculations for these dimensions across different graph classes.

Contribution

It extends switching concepts to higher dimensions and introduces balancing dimensions for better classification of signed graphs' imbalance.

Findings

Defined balancing and strong balancing dimensions.

Provided bounds for these dimensions.

Calculated dimensions for specific graph classes.

Abstract

A signed graph is a graph with edges marked positive and negative; it is unbalanced if some cycle has negative sign product. We introduce the concept of vector valued switching function in signed graphs, which extends the concept of switching to higher dimensions. Using this concept, we define balancing dimension and strong balancing dimension for a signed graph, which can be used for a new classification of degree of imbalance of unbalanced signed graphs. We provide bounds for the balancing and strong balancing dimensions, and calculate these dimensions for some classes of signed graphs.