The Berge–Sauer conjecture (see [2, 3]) says that any simple (no multiple edges and loops) 4-regular graph contains a 3-regular subgraph. This conjecture was proved in [4, 6]. In [1, 2] the Chevalley–Warning theorem was used to extend this result to graphs with multiple edges, which are 4-regular plus an edge. Our main result, Theorem 2.2, presents the sufficient condition for a 4-regular graph...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

Suppose G is a connected, k-regular graph such that Spec…G† ˆ Spec…G† where G is a distance-regular graph of diameter d with parameters a1 ˆ a2 ˆ ˆ adÿ1 ˆ 0 and ad > 0; i.e., a generalized odd graph, we show that G must be distance-regular with the same intersection array as that of G in terms of the notion of Ho ̈man Polynomials. Furthermore, G is isomorphic to G if G is one of the odd polygon ...

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 ...

A subgroup G of automorphisms of a graph X is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be 1 2-transitive if Aut X is 1 2-transitive. The graph X is called one-regular if Aut X acts regularly on the set arcs of X. The interplay of three diierent concepts of maps, one-regular graphs and 1 2-transitive group actions on graphs of valency 4 is...

For example, the pentagon is strongly regular with parameters (v, k, λ, μ) = (5, 2, 0, 1). One easily verifies that a graph G is strongly regular with parameters (v, k, λ, μ) if and only if its complement G is strongly regular with parameters (v, v−k− 1, v− 2k+μ− 2, v−2k+λ). The line graph of the complete graph of order m, known as the triangular graph T (m), is strongly regular with parameters...

A simple graph G is k-ordered (respectively, k-ordered hamiltonian), if for any sequence of k distinct vertices v1, . . . , vk of G there exists a cycle (respectively, hamiltonian cycle) in G containing these k vertices in the specified order. In 1997 Ng and Schultz introduced these concepts of cycle orderability and posed the question of the existence of 3-regular 4-ordered (hamiltonian) graph...

By Seidel’s switching, we construct new strongly regular graphs with parameters (276, 140, 58, 84). In this paper, we simplify the known switching theorem due to Bose and Shrikhande as follows. Let G = (V,E) be a primitive strongly regular graph with parameters (v, k, λ, μ). Let S(G,H) be the graph from G by switching with respect to a nonempty H ⊂ V . Suppose v = 2(k − θ1) where θ1 is the nont...

There is a type of distance-regular graph, said to be $Q$-polynomial. In this paper we investigate generalized $Q$-polynomial property involving graph that not necessarily distance-regular. We give detailed description an example associated with the projective geometry $L_N(q)$.

Given a A:-regular graph G of order n, what is the minimum number v(G) of extra vertices required to embed G in a /<+1-regular graph? Clearly v(G) = 0 precisely when the complement G of G has a 1-factor—in particular, when n ^ 2k is even. Suppose G has no 1-factor: if n, k have opposite parity we show that v(G) = 1, while if n, k have the same parity (which must then be even with n < 2k) we sho...