let $a$ be a non-trivial abelian group and $a^{*}=asetminus {0}$. a graph $g$ is said to be $a$-magic graph if there exists a labeling$l:e(g)rightarrow a^{*}$ such that the induced vertex labeling$l^{+}:v(g)rightarrow a$, define by $$l^+(v)=sum_{uvin e(g)} l(uv)$$ is a constant map.the set of all constant integerssuch that $sum_{uin n(v)} l(uv)=c$, for each $vin n(v)$,where $n(v)$ denotes the s...

The commuting graph of a finite non-abelian group G with center Z(G), denoted by ?c(G), is simple undirected whose vertex set G?Z(G), and two distinct vertices x y are adjacent if only xy=yx. Alwardi et al. (Bulletin, 2011, 36, 49-59) defined the common neighborhood matrix CN(G) energy Ecn(G) G. A called CN-hyperenergetic Ecn(G)>Ecn(Kn), where n=|V(G)| Kn denotes complete on n vertices. Two ...

Let Fλ(S) be the space of tensor densities of degree (or weight) λ on the circle S. The spaceDk λ,μ(S1) of k-th order linear differential operators from Fλ(S) to Fμ(S) is a natural module over Diff(S), the diffeomorphism group of S. We determine the algebra of symmetries of the modules Dk λ,μ(S1), i.e., the linear maps on Dk λ,μ(S1) commuting with the Diff(S)-action. We also solve the same prob...

Let k¿2; n=2+1; and let m0; : : : ; mk−1 each be a multiple of n. The graph Cm0×· · ·×Cmk−1 consists of isomorphic connected components, each of which is (n − 1)-regular and admits of a vertex partition into n smallest independent dominating sets. Accordingly, (independent) domination number of each connected component of this graph is equal to (1=n)th of the number of vertices in it. ? 2001 El...

We consider a problem coming from practical applications: finding a minimum spanning tree with both edge weights and inner node (non-leaf node) weights. This problem is NP-complete even in the metric space. We present two polynomial time algorithms which achieve approximation factors of 2.35 ln n and 2(Hn − 1), respectively, where n is the number of nodes in the graph and Hn is the n-th Harmoni...

Let G be a finite group and X be a union of conjugacy classes of G. Define C(G,X) to be the graph with vertex set X and x, y ∈ X (x 6= y) joined by an edge whenever they commute. In the case that X = G, this graph is named commuting graph of G, denoted by ∆(G). The aim of this paper is to study the automorphism group of the commuting graph. It is proved that Aut(∆(G)) is abelian if and only if ...

The Sylow graph of a finite group originates from recent investigations on certain classes of groups, defined in terms of normalizers of Sylow subgroups. The connectivity of this graph has been proved only last year with the use of the classification of finite simple groups (CFSG). A series of interesting questions arise naturally. First of all, it is not clear whether it is possible to avoid C...

The n-th power (n 1) of a graph G = (V; E), written G n , is deened to be the graph having V as its vertex set with two vertices u; v adjacent in G n if and only if there exists a path of length at most n between them. Similarly, graph H has an n-th root G if G n = H. For the case of n = 2, we say that G 2 is the square of G and G is the square root of G 2. Here we give a linear time algorithm ...

We study a large class of models with an arbitrary (finite) number of degrees of freedom, described by Hamiltonians which are polynomial in bosonic creation and annihilation operators, and including as particular cases n-th harmonic generation and photon cascades. For each model, we construct a complete set of commuting integrals of motion of the Hamiltonian, fully characterize the common eigen...