For k = 2, 3 and a cubic graph G let νk(G) denote the size of a maximum k-edge-colorable subgraph of G. Mkrtchyan, Petrosyan and Vardanyan proved that ν2(G) ≥ 4 5 · |V (G)|, ν3(G) ≥ 7 6 · |V (G)| [13]. They were also able to show that ν2(G) + ν3(G) ≥ 2 · |V (G)| [3] and ν2(G) ≤ |V (G)|+2·ν3(G) 4 [13]. In the present work, we show that the last two inequalities imply the first two of them. Moreo...

BACKGROUND Recent meta-analyses have studied population differences concerning interleukin (IL)-1 gene polymorphisms in gastric carcinogenesis. In addition to the IL-1 gene cluster, candidate genes include those encoding the pro-inflammatory cytokine tumor necrosis factor (TNF)-α. The aim of the study was to systematically review the role of TNF-α-238 and TNF-α-308 gene polymorphisms (genotypes...

A maximum independent set of vertices in a graph is a set of pairwise nonadjacent vertices of largest cardinality α. Plummer [14] defined a graph to be well-covered, if every independent set is contained in a maximum independent set of G. One of the most challenging problems in this area, posed in the survey of Plummer [15], is to find a good characterization of well-covered graphs of girth 4. ...

The square of a graph G is the graph G with the same vertex set as in G, and an edge of G is joining two distinct vertices, whenever the distance between them in G is at most 2. G is a square-stable graph if it enjoys the property α(G) = α(G), where α(G) is the size of a maximum stable set in G. In this paper we show that G is a König-Egerváry graph if and only if G is a square-stable König-Ege...

Let g be a real semisimple Lie algebra and θ a fixed Cartan involution of g. In this paper the subscript C is used for indicating the complexification of a real object. Denote the universal enveloping algebra of the complex Lie algebra gC by U(gC), the center of U(gC) by Z(gC), and the symmetric algebra of gC by S (gC). Similar notation is used for other complex Lie algebras or vector spaces. L...

If G is a triangle-free graph, then two Gallai identities can be written as α(G)+ χ(L(G)) = |V (G)| = α(L(G))+ χ(G), where α and χ denote the stability number and the clique-partition number, and L(G) is the line graph of G. We show that, surprisingly, both equalities can be preserved for any graph G by deleting the edges of the line graph corresponding to simplicial pairs of adjacent arcs, acc...

We show that, in a k-connected graph G of order n with α(G) = α, between any pair of vertices, there exists a path P joining them with |P | ≥ min { n, (k−1)(n−k) α + k } . This implies that, for any edge e ∈ E(G), there is a cycle containing e of length at least min { n, (k−1)(n−k) α + k } . Moreover, we generalize our result as follows: for any choice S of s ≤ k vertices in G, there exists a t...

Weyl denominator identity for the affinization of a basic Lie superalgebra with non-zero Killing form was formulated by V. Kac and M. Wakimoto and was proven by them for the defect one case. In this paper we prove this identity. 0. Introduction Let g be a basic Lie superalgebra with a non-zero Killing form. Let ĝ be the affinization of g. Let h (resp., ĥ) be the Cartan subalgebra in g (resp., i...

For a simple finite graph G, the generalized adjacency matrix is defined as Aα(G)=αD(G)+(1−α)A(G),α∈[0,1], where A(G) and D(G) are respectively diagonal of vertex degrees. The Aα-spread G difference between largest eigenvalue smallest Aα(G). In this paper, we answer question posed in (Lin, Z.; Miao, L.; Guo, S. Bounds on graph. Electron. J. Linear Algebra 2020, 36, 214–227). Furthermore, show t...

Let α be a permutation of the vertex set V (G ) of a connected graph G. Define the total relative displacement of α in G by δα (G )G ∑ x,y∈V(G) dG (x, y)AdG (α (x), α (y)) , where dG (x, y) is the length of the shortest path between x and y in G. Let π*(G ) be the maximum value of δα (G ) among all permutations of V (G ). The permutation which realizes π*(G ) is called a chaotic mapping of G. I...