The rainbowness, rb(G), of a connected plane graph G is the minimum number k such that any colouring of vertices of the graph G using at least k colours involves a face all vertices of which have different colours. For a cubic polyhedral (i.e. 3-connected plane) graph G we prove that n 2 + α∗ 1 − 1 ≤ rb(G) ≤ n− α∗ 0 + 1 , where α∗ 0 and α ∗ 1 denote the independence number and the edge independ...

Let G be a connected bipartite graph. An involution α of G that preserves the bipartition of G is called bipartite. Let Gα be the graph obtained from G by adding to G the natural perfect matching induced by α. We show that the k-cube Qk is isomorphic to the direct product G×H if and only if G is isomorphic to Qk−1 for some bipartite involution α of Qk−1 and H = K2.

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx α, α = α (G) , is called the independence polynomial of G (I. Gutman and F. Harary, 1983). If sα−i = f (i) · si holds for every i ∈ {0, 1, ..., ⌊α/2⌋}, t...

In this paper we determine the local and global resilience of random graphs Gn,p (p n−1) with respect to the property of containing a cycle of length at least (1 − α)n. Roughly speaking, given α > 0, we determine the smallest rg(G,α) with the property that almost surely every subgraph of G = Gn,p having more than rg(G,α)|E(G)| edges contains a cycle of length at least (1−α)n (global resilience)...

Let α ∈ (0, 1) and let G = (VG, EG) be a graph. According to Dunbar, Hoffman, Laskar and Markus [3] a set D ⊆ VG is called an α-dominating set of G, if |NG(u)∩D| ≥ αdG(u) for all u ∈ VG \D. We prove a series of upper bounds on the α-domination number of a graph G defined as the minimum cardinality of an α-dominating set of G.

The stability number α(G) of the graph G is the size of a maximum stable set of G. If sk denotes the number of stable sets of cardinality k in graph G, then I(G;x) = α(G)

For α≥1, β≥0, and a graph G, spanning subgraph H of G is said to be an (α,β)-spanner if dist(u,v,H)≤α⋅dist(u,v,G)+β holds for any pair vertices u v. These type spanners, called linear generalizes additive spanners multiplicative spanners. Recently, Fomin, Golovach, Lochet, Misra, Saurabh, Sharma initiated the study directed graphs (IPEC 2020). In this article, we continue line research prove th...

Let G be a simple undirected graph on n vertices. V. Nikiforov studied hybrids of AG and DG defined the matrix AαG for every real α∈[0,1] as AαG=αDG+(1−α)AG. In this paper, we define generalized Randić G, introduce establish bounds Estrada index new matrix. Furthermore, find smallest value α which is positive semidefinite. Finally, present solution to problem proposed by Nikiforov. The consists...

The present study aimed to assess the effect of polymorphisms in the tumor necrosis factor α (TNF-α) promoter (A/A, A/G and G/G) and exons (T/T, T/C and C/C) on immune function and reproductive performance in dairy cows. The occurrence of the first postpartum ovulation within 3 weeks in the cows with the TNF-α promoter A/G and G/G genotypes was higher than in the A/A group. Among the different ...

Since χ(G) · α(G) ≥ |V (G)|, Hadwiger’s Conjecture implies that any graph G has the complete graph Kdn α e as a minor, where n is the number of vertices of G and α is the maximum number of independent vertices in G. Motivated by this fact, it is shown that any graph on n vertices with independence number α ≥ 3 has the complete graph Kd n 2α−2 e as a minor. This improves the well-known theorem o...