On the degree pairs of a graph (original) (raw)

On edge-regular graphs with kge3b_1−3k\ge 3 b_1-3kge3b_13

St. Petersburg Mathematical Journal, 2000

An undirected graph on v vertices in which the degrees of all vertices are equal to k and each edge belongs to exactly λ triangles is said to be edge-regular with parameters (v, k, λ). It is proved that an edge-regular graph with parameters (v, k, λ) such that k ≥ 3b 1 − 3 either has diameter 2 and coincides with the graph P (2) on 20 vertices or with the graph M (19) on 19 vertices; or has at most 2k + 4 vertices; or has diameter at least 3 and is a trivalent graph without triangles, or the line graph of a quadrivalent graph without triangles, or a locally hexagonal graph; or has diameter 3 and satisfies |Γ 3 (u)| ≤ 1 for each vertex u.

On the total domatic number of regular graphs

2012

A set S of vertices of a graph G = (V, E) without isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domatic number of a graph G is the maximum number of total dominating sets into which the vertex set of G can be partitioned. We show that the total domatic number of a random r-regular graph is almost surely at most r − 1, and that for 3-regular random graphs, the total domatic number is almost surely equal to 2. We also give a lower bound on the total domatic number of a graph in terms of order, minimum degree and maximum degree. As a corollary, we obtain the result that the total domatic number of an r-regular graph is at least r/(3 ln(r)). MSC(2010): 05C69.

IJERT-Some (r, 2, k)-regular graphs containing a given graph

International Journal of Engineering Research and Technology (IJERT), 2012

https://www.ijert.org/some-r-2-k-regular-graphs-containing-a-given-graph https://www.ijert.org/research/some-r-2-k-regular-graphs-containing-a-given-graph-IJERTV1IS10467.pdf A graph G is called (r, 2 , k)-regular if each vertex of G is at a distance one from r vertices of G and each vertex of G is at a distance two from exactly k vertices of G. That is , if d(v) = r and d 2 (v) = k , for all v in G [9].This paper suggests a method to construct a (m + n-1, 2, (m n-1))-regular graph of order m n containing the given graph G of order n ≥ 2 as an induced sub graph, for any m ≥ 1, and this paper includes existence of some (r , 2 , k)-regular graphs and few examples of (2 , k) regular graphs.

The Domatic Number of Regular Graphs

Ars Combinatoria - ARSCOM, 2004

The domatic number of a graph G is the maximum number of dominating sets into which the vertex set of G can be partitioned. We show that the domatic number of a random r-regular graph is almost surely at most r, and that for 3-regular random graphs, the domatic number is almost surely equal to 3. We also give a lower bound on the domatic number of a graph in terms of order, minimum degree and maximum degree. As a corol- lary, we obtain the result that the domatic number of an r-regular graph is at least (r + 1)/(3ln(r + 1)).

The regular number of a graph

Journal of Discrete Mathematical Sciences and Cryptography, 2012

Let G be a simple undirected graph. The regular number of G is defined to be the minimum number of subsets into which the edge set of G can be partitioned so that the subgraph induced by each subset is regular. In this work, we obtain the regular number of some families of graphs and discuss some general bounds on this parameter. Also, some of the lower or upper bounds proved in [4] are shown here to hold with equality. Key words-Generalized graph colorings, regular number of a graph, the degree bound, minimum regular partition of the edge set.

Large regular graphs with no induced 2K 2

Graphs and Combinatorics, 1992

Let r be a positive integer. Consider r-regular graphs in which no induced subgraph on four vertices is an independent pair of edges. The number v of vertices in such a graph does not exceed 5r/2; this proves a conjecture of Bermond. More generally, it is conjectured that if v > 2r, then the ratio v/r must be a rational number of the form 2 + 1/(2k). This is proved for v/r > 21 _~. The extremal graphs and many other classes of these graphs are described and characterized.

A family of regular graphs of girth 5

Discrete Mathematics, 2008

Murty [A generalization of the Hoffman-Singleton graph, Ars Combin. 7 (1979) 191-193.] constructed a family of (p m + 2)regular graphs of girth five and order 2p 2m , where p 5 is a prime, which includes the Hoffman-Singleton graph [A.J. Hoffman, R.R. Singleton, On Moore graphs with diameters 2 and 3, IBM J. (1960) 497-504]. This construction gives an upper bound for the least number f (k) of vertices of a k-regular graph with girth 5. In this paper, we extend the Murty construction to k-regular graphs with girth 5, for each k. In particular, we obtain new upper bounds for f (k), k 16.