Linear programming bounds for regular graphs

TL;DR

This paper extends the linear programming method to derive bounds on the number of vertices in regular graphs with specified eigenvalues, providing new insights into graph eigenvalue properties and their relation to girth.

Contribution

It develops a novel linear programming approach for regular graphs based on association schemes, linking eigenvalues, girth, and graph structure.

Findings

Connected k-regular graphs with girth > 2d-1 minimize the second-largest eigenvalue.

Identifies classes of graphs satisfying the girth and eigenvalue conditions, including Moore graphs and generalized polygons.

Provides bounds that characterize extremal regular graphs based on spectral and girth properties.

Abstract

Delsarte, Goethals, and Seidel (1977) used the linear programming method in order to find bounds for the size of spherical codes endowed with prescribed inner products between distinct points in the code. In this paper, we develop the linear programming method to obtain bounds for the number of vertices of connected regular graphs endowed with given distinct eigenvalues. This method is proved by some "dual" technique of the spherical case, motivated from the theory of association scheme. As an application of this bound, we prove that a connected $k$-regular graph satisfying $g>2d-1$ has the minimum second-largest eigenvalue of all $k$-regular graphs of the same size, where $d$ is the number of distinct non-trivial eigenvalues, and $g$ is the girth. The known graphs satisfying $g>2d-1$ are Moore graphs, incidence graphs of regular generalized polygons of order $(s,s)$, triangle-free strongly regular graphs, and the odd graph of degree $4$.