in this paper, it was shown that , where  and , and , where  is not prime and , are od-characterizable.

the wiener index w(g) of a connected graph g is defined as the sum of the distances betweenall unordered pairs of vertices of g. the eccentricity of a vertex v in g is the distance to avertex farthest from v. in this paper we obtain the wiener index of a graph in terms ofeccentricities. further we extend these results to the self-centered graphs.

Abst rac t . We survey and present new geometric and combinatorial propertiez of some polyhedra with application in combinatorial optimization, for example, the max-cut and multicommodity flow problems. Namely we consider the volume, symmetry group, facets, vertices, face lattice, diameter, adjacency and incidence relm :ons and connectivity of the metric polytope and its relatives. In partic~da...

the vertex arboricity $rho(g)$ of a graph $g$ is the minimum number of subsets into which the vertex set $v(g)$ can be partitioned so that each subset induces an acyclic graph‎. ‎a graph $g$ is called list vertex $k$-arborable if for any set $l(v)$ of cardinality at least $k$ at each vertex $v$ of $g$‎, ‎one can choose a color for each $v$ from its list $l(v)$ so that the subgraph induced by ev...

We study the spectral radius of graphs with n vertices and k cut edges. In this paper, we show that of all the connected graphs with n vertices and k cut edges, the maximal spectral radius is obtained uniquely at Kk n , where K k n is a graph obtained by joining k independent vertices to one vertex of Kn−k . We also discuss the limit point of the maximal spectral radius. © 2004 Elsevier Inc. Al...

Let G be a simple connected graph. The transmission of any vertex v of a graph G is defined as the sum of distances of a vertex v from all other vertices in a graph G. Then the distance signless Laplacian matrix of G is defined as D^{Q}(G)=D(G)+Tr(G), where D(G) denotes the distance matrix of graphs and Tr(G) is the diagonal matrix of vertex transmissions of G. For a given minimum dominating se...

Abstract Let G be a finite group. Denoting by $$\textrm{cd}(G)$$ cd ( G ) the set of degrees irreducible complex characters , we consider character degree graph : this is (simple undirected) whose vertices are prime divisors numbe...

a roman dominating function (rdf) on a graph g = (v,e) is defined to be a function satisfying the condition that every vertex u for which f(u) = 0 is adjacent to at least one vertex v for which f(v) = 2. a set s v is a restrained dominating set if every vertex not in s is adjacent to a vertex in s and to a vertex in . we define a restrained roman dominating function on a graph g = (v,e) to be ...

Given a graph G = (V; E), the metric polytope S(G) is deened by the inequalities x(F) ? x(C n F) jFj ? 1 for F C; jFj odd ; C cycle of G, and 0 x e 1 for e 2 E. Optimization over S(G) provides an approximation for the max-cut problem. The graph G is called 1 d-integral if all the vertices of S(G) have their coordinates in f i d j 0 i dg. We prove that the class of 1 d-integral graphs is closed ...

For any finite abelian group G and any subset S ⊆ G, we determine the connectivity of the addition Cayley graph induced by S on G. Moreover, we show that if this graph is not complete, then it possesses a minimum vertex cut of a special, explicitly described form.