Unique Metro Domination Number of Circulant Graphs

In this paper, we determine unique metro domination number of circulant graphs

On Classes of Neighborhood Resolving Sets of a Graph

Let G=(V,E) be a simple connected graph. A subset S of V is called a neighbourhood set of G if G=\bigcup_{s\in S}<N[s]>, where N[v] denotes the closed neighbourhood of the vertex v in G. Further for each ordered subset S={s_1,s_2, ...,s_k} of V and a vertex $u\in V$, we associate a vector $\Gamma(u/S)=(d(u,s_1),d(u,s_2), ...,d(u,s_k))$ with respect to S, where d(u,v) denote the distance between u and v in G. A subset S is said to be resolving set of G if $\Gamma(u/S)\neq \Gamma(v/S)$ for all $u,v\in V-S$. A neighbouring set of G which is also a resolving set for G is called a neighbourhood resolving set (nr-set). The purpose of this paper is to introduce various types of nr-sets and compute minimum cardinality of each set, in possible cases, particulary for paths and cycles

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded

Proteomic Interrogation of Androgen Action in Prostate Cancer Cells Reveals Roles of Aminoacyl tRNA Synthetases

Prostate cancer remains the most common malignancy among men in United States, and there is no remedy currently available for the advanced stage hormone-refractory cancer. This is partly due to the incomplete understanding of androgen-regulated proteins and their encoded functions. Whole-cell proteomes of androgen-starved and androgen-treated LNCaP cells were analyzed by semi-quantitative MudPIT ESI- ion trap MS/MS and quantitative iTRAQ MALDI- TOF MS/MS platforms, with identification of more than 1300 high-confidence proteins. An enrichment-based pathway mapping of the androgen-regulated proteomic data sets revealed a significant dysregulation of aminoacyl tRNA synthetases, indicating an increase in protein biosynthesis- a hallmark during prostate cancer progression. This observation is supported by immunoblot and transcript data from LNCaP cells, and prostate cancer tissue. Thus, data derived from multiple proteomics platforms and transcript data coupled with informatics analysis provides a deeper insight into the functional consequences of androgen action in prostate cancer

The polycomb group protein EZH2 is involved in progression of prostate cancer

Prostate cancer is a leading cause of cancer-related death in males and is second only to lung cancer. Although effective surgical and radiation treatments exist for clinically localized prostate cancer, metastatic prostate cancer remains essentially incurable. Here we show, through gene expression profiling(1), that the polycomb group protein enhancer of zeste homolog 2 (EZH2)(2,3) is overexpressed in hormone-refractory, metastatic prostate cancer. Small interfering RNA (siRNA) duplexes(4) targeted against EZH2 reduce the amounts of EZH2 protein present in prostate cells and also inhibit cell proliferation in vitro. Ectopic expression of EZH2 in prostate cells induces transcriptional repression of a specific cohort of genes. Gene silencing mediated by EZH2 requires the SET domain and is attenuated by inhibiting histone deacetylase activity. Amounts of both EZH2 messenger RNA and EZH2 protein are increased in metastatic prostate cancer; in addition, clinically localized prostate cancers that express higher concentrations of EZH2 show a poorer prognosis. Thus, dysregulated expression of EZH2 may be involved in the progression of prostate cancer, as well as being a marker that distinguishes indolent prostate cancer from those at risk of lethal progression.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/62896/1/nature01075.pd

Strong and Semi Strong Outer Mod Sum Graphs

A semi strong outer mod sum labeling of a graph G is an injective mapping f: V (G) → Z + with an additional property that for each vertex v of G, there exist vertices w1, w2 in V (G) such that f(w1) = ∑ u∈N(v) f(u) and f(v) =∑u∈N(w2) f(u), where both the sums are taken under addition modulo m for some positive integer m. A graph G which admits a semi strong outer mod sum labeling is called a semi strong outer mod sum graph. In this paper we show that the Paths Pn for all n = 3, the Cycle C6, and the Complete graphs Kn for all n = 3 are semi strong outer mod sum graphs

Radio Number of Cube of a Path

Abstract: Let G be a connected graph. For any two vertices u and v, let d(u, v) denotes the distance between u and v in G. The maximum distance between any pair of vertices is called the diameter of G and is denoted by diam(G). A Smarandachely k-radio labeling of a connected graph G is an assignment of distinct positive integers to the vertices of G, with x ∈ V (G) labeled f(x), such that d(u, v) + |f(u) − f(v) | ≥ k + diam(G). Particularly, if k = 1, such a Smarandachely radio k-labeling is called radio labeling for abbreviation. The radio number rn(f) of a radio labeling f of G is the maximum label assignment to a vertex of G. The radio number rn(G) of G is minimum {rn(f)} over all radio labelings of G. In this paper, we completely determine the radio number of the graph P 3 n for all n ≥ 4. Keywords: Smarandachely radio k-labeling, radio labeling, radio number of a graph

Smarandachely k-Constrained Number of Paths and Cycles

Abstract: A Smarandachely k-constrained labeling of a graph G(V, E) is a bijective mapping f: V ∪ E → {1, 2,.., |V | + |E|} with the additional conditions that |f(u) − f(v) | ≥ k whenever uv ∈ E, |f(u)−f(uv) | ≥ k and |f(uv)−f(vw) | ≥ k whenever u ̸ = w, for an integer k ≥ 2. A graph G which admits a such labeling is called a Smarandachely k-constrained total graph, abbreviated as k −CTG. The minimum number of isolated vertices required for a given graph G to make the resultant graph a k − CTG is called the k-constrained number of the graph G and is denoted by tk(G). In this paper we settle the open problems 3.4 and 3.6 in [4] by showing that tk(Pn) = 0, if k ≤ k0; 2(k − k0), if k&gt; k0 and 2n ≡ 1 or 2 (mod 3); 2(k − k0) − 1 if k&gt; k0; 2n ≡ 0(mod 3) and tk(Cn) = 0, if k ≤ k0; 2(k − k0), if k&gt; k0 and 2n ≡ 0 (mod 3); 3(k − k0) if k&gt; k0 and 2n ≡ 1 or 2 (mod 3), where k0 = ⌊ 2n−