In this paper, we construct intriguing sets in five classes of strongly regular graphs defined on nonisotropic points finite classical polar spaces, and determine their intersection numbers.

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices u and v of G, a rainbow u− v geodesic in G is a rainbow u− v path of length d(u, v), where d(u, v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow u − v geodesic for any two vertices...

Confirming a conjecture by Ivančo and Jendrol’ for a large class of graphs we prove that for every graph G = (V,E) of order n, size m and maximum degree ∆ with m > 111000∆ there is a function f : V ∪ E → { 1, 2, ..., ⌈ m+2 3 ⌉} such that f(u) + f(uv) + f(v) 6= f(u′) + f(u′v′) + f(v′) for every uv, u′v′ ∈ E with uv 6= u′v′. Furthermore, we prove the existence of such a function with values up to...

A graph is locally irregular if every two adjacent vertices have distinct degrees. Recently, Baudon et al. introduced the notion of decomposition into locally irregular subgraphs. They conjectured that for almost every graphG, there exists a minimum integer χirr(G) such thatG admits an edge-partition into χ ′ irr(G) classes, each of which induces a locally irregular graph. In particular, they c...

An irregular assignment is a k -labeling of the edges : {1, 2, , } f E k → ... such that the vertex weights (label sums of edges incident with the vertex) are different for all vertices of G . The smallest k for which there is an irregular assignment is the irregularity strength. The notion of irregularity strength was introduced by Chartrand et al. [8] and studied by numerous authors, see [6,1...

The concepts of covering and matching in fuzzy graphs using strong arcs are introduced and obtained the relationship between them analogous to Gallai’s results in graphs. The notion of paired domination in fuzzy graphs using strong arcs is also studied. The strong paired domination number γspr of complete fuzzy graph and complete bipartite fuzzy graph is determined and obtained bounds for the s...

Let G be a connected, simple, and undirrected graph, where V (G) is the vertex set E(G) edge set. k natural numbers. For graph we define total k−labeling ρ such that vertices of are labeled with {0, 2, 4, . , 2kv} edges {1, 3, ke}, = max{2kv, ke}. Total called an irregular reflexive k− labeling if every two distinct have weights, weight defined as sum label incident to this edge. The minimum ha...

In this paper, we investigate some basic problems related to the strong connectivity and to the 2-connectivity of a directed graph, by considering the effect of edge and vertex deletions on its strongly connected components. Let G be a directed graph with m edges and n vertices. We present a collection of O(n)-space data structures that, after O(m+ n)-time preprocessing can accomplish the follo...

Directed graphs are widely used in modelling of nonsymmetric relations in various sciences and engineering disciplines. We discuss invariants of strongly connected directed graphs minimal number of vertices or edges necessary to remove to make remaining graphs not strongly connected. By analogy with undirected graphs these invariants are called strong vertex/edge connectivities. We review first...

for two normal edge-transitive cayley graphs on groups h and k which have no common direct factor and gcd(jh=h ′j; jz(k)j) = 1 = gcd(jk=k ′j; jz(h)j), we consider four standard products of them and it is proved that only tensor product of factors can be normal edge-transitive.