‎in this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is cohen-macaulay‎. ‎it is proved that if there exists a cover of an $r$-partite cohen-macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

‎In this paper‎, ‎we give some necessary conditions for an $r$-partite graph such that the edge ring of the graph is Cohen-Macaulay‎. ‎It is proved that if there exists a cover of an $r$-partite Cohen-Macaulay graph by disjoint cliques of size $r$‎, ‎then such a cover is unique‎.

in this paper we study the coprime graph of a group $g$. the coprime graph of a group $g$, is a graph whose vertices are elements of $g$ and two distinct vertices $x$ and $y$ are adjacent iff $(|x|,|y|)=1$. in this paper we classify all the groups which the coprime graph is a complete r-partite graph or a planar graph. also we study the automorphism group of the coprime graph.

Let R be a commutative ring and G(R) be a graph with vertices as proper andnon-trivial ideals of R. Two distinct vertices I and J are said to be adjacentif and only if I + J = R. In this paper we study a graph constructed froma subgraph G(R)Δ(R) of G(R) which consists of all ideals I of R such thatI Δ J(R), where J(R) denotes the Jacobson radical of R. In this paper westudy about the relation b...

An [n, k, r]-partite graph is a graph whose vertex set, V , can be partitioned into n pairwisedisjoint independent sets, V1, . . . , Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 ≤ i < j ≤ n. An independent transversal in an [n, k, r]-partite graph is an independent set, T , consisting of n vertices, one from each Vi. An ...

We show that if r s 2; n > r8; and G is a graph of order n containing as many r-cliques as the r-partite Turán graph of order n; then G has more than nr 1=r2r+12 cliques sharing a common edge unless G is isomorphic to the r-partite Turán graph of order n. This structural result generalizes a previous result that has been useful in extremal graph theory.

Let r ≥ 2 and write K+ r (s1, . . . , sr) for the complete r-partite graph with parts of size s1 ≥ 2, . . . , sr with an edge added to the first part. Letting tr (n) be the number of edges of the r-partite Turán graph of order n, we prove the following theorems: (A) For all r ≥ 2, sufficiently small ε > 0 and sufficiently large n, every graph of order n with tr (n) + 1 edges contains a K + r ( ...

In the following material, we use the notion of an independent set of a graph G: this is a set of I vertices of G such that no two vertices of I are adjacent in G. These are sometimes referred to as stable sets in the literature. The chromatic number of a graph G, denoted χ(G), is the smallest r such that V (G) has a partition into independent sets V1, V2, . . . , Vr. In other words, we may ass...

Let μ (G) be the largest eigenvalue of a graph G and Tr (n) be the r-partite Turán graph of order n. We prove that if G is a graph of order n with μ (G) > μ (Tr (n)) , then G contains various large supergraphs of the complete graph of order r + 1, e.g., the complete r-partite graph with all parts of size log n with an edge added to the first part. We also give corresponding stability results.