Girth-regular graphs

Regular Graphs of given Girth

Graph theory is the study of mathematical structures called graphs. We define a graph as a pair (V,E), where V is a nonempty set, and E is a set of unordered pairs of elements of V . V is called the set of vertices of G, and E is the set of edges. Two vertices a and b are adjacent provided (a, b) ∈ E. If a pair of vertices is adjacent, the vertices are referred to as neighbors. We can represent...

Small Regular Graphs of Girth 7

In this paper, we construct new infinite families of regular graphs of girth 7 of smallest order known so far. Our constructions are based on combinatorial and geometric properties of (q + 1, 8)-cages, for q a prime power. We remove vertices from such cages and add matchings among the vertices of minimum degree to achieve regularity in the new graphs. We obtain (q + 1)-regular graphs of girth 7...

On (minimal) regular graphs of girth 6

The second eigenvalue of regular graphs of given girth

Lower bounds on the subdominant eigenvalue of regular graphs of given girth are derived. Our approach is to approximate the discrete spectrum of a finite regular graph by the continuous spectrum of an infinite regular tree. We interpret these spectra as probability distributions and the girth condition as equalities between the moments of these distributions. Then the associated orthogonal poly...

A construction of small regular bipartite graphs of girth 8

Let q be a prime a power and k an integer such that 3 ≤ k ≤ q. In this paper we present a method using Latin squares to construct adjacency matrices of k-regular bipartite graphs of girth 8 on 2(kq − q) vertices. Some of these graphs have the smallest number of vertices among the known regular graphs with girth 8.