Let G =( V, E) be a graph. A bijectionf: V+{ 1,2,. ., 1 VI} IS called a prime labelling if for each e = {u, u} in E, we have GCD(f(u),f(u))= 1. A graph admits a prime labelling is called a prime graph. Around ten years ago, Roger Entringer conjectured that every tree is prime. So far, this conjecture is still unsolved. In this paper, we show that the conjecture is true for trees of order up to ...

let g be a finite group and let $gk(g)$ be the prime graph of g. we assume that $n$ is an odd number. in this paper, we show that if $gk(g)=gk(b_n(p))$, where $ngeq 9$ and $pin {3,5,7}$, then g has a unique nonabelian composition factor isomorphic to $b_n(p)$ or $c_n(p)$ . as consequences of our result, $b_n(p)$ is quasirecognizable by its spectrum and also by a new proof, the validity of a con...

In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph cam be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertextransitive graphs, including Cayley graphs, multidimensional circulants, and...

The Hadwiger number (G) of a graph G is the largest integer h such that the complete graph on h nodes Kh is a minor of G. Equivalently, (G) is the largest integer such that any graph on at most (G) nodes is a minor ofG. The Hadwiger’s conjecture states that for any graph G, (G) (G), where (G) is the chromatic number of G. It is well-known that for any connected undirected graph G, there exists ...

It has long been known that the class of connected nonbipartite graphs (with loops allowed) obeys unique prime factorization over the direct product of graphs. Moreover, it is known that prime factorization is not necessarily unique in the class of connected bipartite graphs. But any prime factorization of a connected bipartite graph has exactly one bipartite factor. Moreover, empirical evidenc...

A graph G is a prime distance graph (respectively, a 2-odd graph) if its vertices can be labeled with distinct integers such that for any two adjacent vertices, the difference of their labels is prime (either 2 or odd). We prove that trees, cycles, and bipartite graphs are prime distance graphs, and that Dutch windmill graphs and paper mill graphs are prime distance graphs if and only if the Tw...

the order graph of a group $g$, denoted by $gamma^*(g)$, is a graph whose vertices are subgroups of $g$ and two distinct vertices $h$ and $k$ are adjacent if and only if $|h|big{|}|k|$ or $|k|big{|}|h|$. in this paper, we study the connectivity and diameter of this  graph. also we give a relation between the order graph and prime  graph of a group.

Tait’s flyping conjecture, stating that two reduced, alternating, prime link diagrams can be connected by a finite sequence of flypes, is extended to reduced, alternating, prime diagrams of 4-regular graphs in S. The proof of this version of the flyping conjecture is based on the fact that the equivalence classes with respect to ambient isotopy and rigid vertex isotopy of graph embeddings are i...

Prime graph of a ring R is a graph whose vertex set is the whole set R any any two elements $x$ and $y$ of $R$ are adjacent in the graph if and only if $xRy = 0$ or $yRx = 0$.  Prime graph of a ring is denoted by $PG(R)$. Directed prime graphs for non-commutative rings and connectivity in the graph are studied in the present paper. The diameter and girth of this graph are also studied in the pa...

Throughout this paper, every groups are finite. The prime graph of a group $G$ is denoted by $Gamma(G)$. Also $G$ is called recognizable by prime graph if for every finite group $H$ with $Gamma(H) = Gamma(G)$, we conclude that $Gcong H$. Until now, it is proved that if $k$ is an odd number and $p$ is an odd prime number, then $PGL(2,p^k)$ is recognizable by prime graph. So if $k$ is even, the r...