A graph G is k-ordered if for any sequence of k distinct vertices v1, v2, . . . , vk of G there exists a cycle in G containing these k vertices in the specified order. In 1997, Ng and Schultz posed the question of the existence of 4-ordered 3-regular graphs other than the complete graph K4 and the complete bipartite graph K3,3. In 2008, Meszaros solved the question by proving that the Petersen ...

The values t = 1, 3, 5, 6, 9 satisfy the standard necessary conditions for existence of a generalised quadrangle of order (3, t). This gives the following possible parameter sets for strongly regular graphs that are pseudo-geometric for such a generalised quadrangle: (v, k, λ, μ) = (16, 6, 2, 2), (40, 12, 2, 4), (64, 18, 2, 6), (76, 31, 2, 7) and (112, 30, 2, 10). It is well-known that there ar...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...