We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

Suppose a graph G have n vertices, m edges, and t triangles. Letting λn (G) be the largest eigenvalue of the Laplacian of G and μn (G) be the smallest eigenvalue of its adjacency matrix, we prove that λn (G) ≥ 2m2 − 3nt m (n2 − 2m) , μn (G) ≤ 3n3t− 4m3 nm (n2 − 2m) , with equality if and only if G is a regular complete multipartite graph. Moreover, if G is Kr+1-free, then λn (G) ≥ 2mn (r − 1) (...

The degree Ramsey number of a graph G, denoted R∆(G; s), is min{∆(H) : H s → G}, where H s → G means that every s-edge-coloring of H contains a monochromatic copy of G. The closed k-blowup of a graph is obtained by replacing every vertex with a clique of size k and every edge with a complete bipartite graph where both partite sets have size k. We prove that there is a function f such that R∆(G;...

Quantifying genuine entanglement is a crucial task in quantum information theory.In this work, we give an approach of constituting $m$-partite measures from any bipartite and $k$-partite measure, $3\leq k<m$. In addition, as complement to the three-qubit concurrence triangle proposed [Phys. Rev. Lett., 127, 040403], show that relation also valid for continuous measure system with dimension. We ...

Let F be a fixed graph of chromatic number r + 1. We prove that for all large n the degree sequence of any F -free graph of order n is, in a sense, close to being dominated by the degree sequence of some r-partite graph. We present two different proofs: one goes via the Regularity Lemma and the other uses a more direct counting argument. Although the latter proof is longer, it gives better esti...

A graph $G$ is called a complete $r$-partite $(r\geq 2)$ graph, if its vertices can be divided into $r$ non-empty independent sets $V_1,\ldots,V_r$ in way that each vertex $V_i$ adjacent to all the other $V_j$ for $1\leq i&lt;j\leq r$. Let $K_{n_{1},n_{2},\ldots,n_{r}}$ denote with $V_1,V_2,\ldots,V_r$ of sizes $n_{1},n_{2},\ldots,n_{r}$. An edge-coloring colors $1,2,\ldots,t$ an \emph{interval...

A digraph D is called is semicomplete c-partite if its vertex set V (D) can be partitioned into c sets (partite sets) such that for any two vertices x and y in diierent partite sets at least one arc between x and y is in D and there are no arcs between vertices in the same partite set. The path covering number of D is the minimum number of paths in D that are pairwise vertex disjoint and cover ...

A multipartite or c-partite tournament is an orientation of a complete c-partite graph. Lu and Guo (submitted for publication) [3] recently introduced strong quasi-Hamiltonianconnectivity of a multipartite tournament D as follows: For any two distinct vertices x and y ofD, there is a pathwith at least one vertex from each partite set ofD from x to y and from y to x. We obtain the definition for...

As usual λ(G) denotes the edge-connectivity of graph G. It was shown in [2] that every G contains a spanning (λ(G)+1)-partite subgraph H such λ(H)=λ(G) and one can find polynomial time. We determine complexity deciding, for given positive integers r,k whether r-colourable which is k-edge-connected. show problem polynomially solvable when r>k NP-complete otherwise. In fact, combined with result ...

In this paper, it is shown that the graph obtained by overlapping the cycle and the complete tripartite graph at an edge is uniquely determined by its chromatic polynomial. ) 3 ( ≥ m Cm 2 , 2 , 2 K Let G be a finite graph with neither loops nor multiple edges and let ) ; ( λ G P denote its chromatic polynomial. Then G is said to be chromatically unique if ) ; ( ) ; ( λ λ G Y P = implies that Y ...