It is better to be semi-regular when you have a low degree

TL;DR

This paper analyzes the algebraic connectivity of random semi-regular graphs, revealing conditions under which they outperform regular graphs and introducing a generalization to non-integer degrees.

Contribution

It provides explicit spectral analysis of semi-regular graphs, compares their connectivity to regular graphs, and introduces a non-integer degree generalization with polynomial characterization.

Findings

Semi-regular bipartite graphs can have higher algebraic connectivity than regular graphs with same vertices and edges for certain degrees.

Regular graphs outperform semi-regular graphs when degree $d\,\geq 8$.

Constructed a small-world network with average degree 2.5 and high algebraic connectivity.

Abstract

We study the algebraic connectivity for several classes of random semi-regular graphs. For large random semi-regular bipartite graphs, we explicitly compute both their algebraic connectivity and as well as the full spectrum distribution. For an integer $d\in\left[ 3,7\right] $, we find families of random semi-regular graphs that have higher algebraic connectivity than a random $d$-regular graphs with the same number of vertices and edges. On the other hand, we show that regular graphs beat semi-regular graphs when $d\geq8.$ More generally, we study random semi-regular graphs whose average degree is $d$, not necessary an integer. This provides a natural generalization of a $d$-regular graph in the case of a non-integer $d.$ We characterise their algebraic connectivity in terms of a root of a certain 6th-degree polynomial. Finally, we construct a small-world-type network of average degree 2.5 with a relatively high algebraic connectivity. We also propose some related open problems and conjectures.