Search for minimal trivalent cycle permutation graphs with girth nine

The Smallest Cubic Graphs of Girth Nine

We describe two computational methods for the construction of cubic graphs with given girth. These were used to produce two independent proofs that the (3, 9)-cages, defined as the smallest cubic graphs of girth 9, have 58 vertices. There are exactly 18 such graphs. We also show that cubic graphs of girth 11 must have at least 106 vertices and cubic graphs of girth 13 must have at least 196 ver...

On (minimal) regular graphs of girth 6

The Structure of Trivalent Graphs with Minimal Eigenvalue Gap

Let G be a connected trivalent graph on n vertices (n ≥ 10) such that among all connected trivalent graphs on n vertices, G has the largest possible second eigenvalue. We show that G must be reduced path-like, i.e. G must be of the form: where the ends are one of the following: q q q q q @ @ @ @ @ or q q @ @ @ q q q q @ @q (the right-hand end block is the mirror image of one of the blocks shown...

Characterization of Trivalent Graphs with Minimal Eigenvalue Gap*

Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conject...

Algorithmic Search for Extremal Graphs of Girth At Least Five

Let f(v) denote the maximum number of edges in a graph of order v and of girth at least 5. In this paper, we discuss algorithms for constructing such extremal graphs. This gives constructive lower bounds of f(v) for v ≤ 200. We also provide the exact values of f(v) for v ≤ 24, and enumerate the extremal graphs for v ≤ 10.