background: b re a st con s er v i n g surg e ry ( b c s ) f o l l owed b y loc a l rad i ot h e ra p y is the standard t r eatment for e a rly sta g e of breast cancer. margin stat u s may have an important impact on local recurren c e and over a ll s u rvival of patients treated with bcs, but this h a s not b een defined adequately. in this study, we in ve stigated the rela t io n ship betwee...

Given a graph G with vertex set V (G) = V and edge set E(G) = E, let Gl be the line graph and Gc the complement of G. Let G0 be the graph with V (G0) = V and with no edges, G1 the complete graph with the vertex set V , G+ = G and G− = Gc. Let B(G) (Bc(G)) be the graph with the vertex set V ∪E and such that (v, e) is an edge in B(G) (resp., in Bc(G)) if and only if v ∈ V , e ∈ E and vertex v is ...

All graphs considered here are finite simple graphs, i.e., graphs without loops, multiple edges or directed edges. For a graph G = (V, E), where V is a vertex set and E is an edge set, we write sometimes V (G) for V and E(G) for E to avoid ambiguity. We shall write G \ v instead of G V \{v} = (V \ {v}, E ∩ 2 V \{v}), the subgraph induced by V \ {v}. A vertex v ∈ V (G) is called a cut-vertex of ...

Let V be a finite non-void set ; V lary. Every finite sequence of elements in ia called a vocnbuV is said to be a s t r i n g o n V. G i v e n a s t r i n g x = a l a 2 . . . a n , t h e n u m b e r n i s c a l l e d t h e l e n g t h o f x . The s t r i n g o f l e n g t h z e r o i s c a l l e d t h e n t r i n g a n d i s d e n o t e d b y r~J . Any s e t o f s t r i n g s o n V i s c a l l ...

In this paper we introduce the notion of minimum-weight edge-discriminators in hypergraphs, and study their various properties. For a hypergraph H = (V,E ), a function λ : V → Z+∪{0} is said to be an edge-discriminator onH if∑v∈Ei λ(v) > 0, for all hyperedges Ei ∈ E , and ∑ v∈Ei λ(v) 6= ∑ v∈Ej λ(v), for every two distinct hyperedges Ei, Ej ∈ E . An optimal edge-discriminator on H, to be denoted...

let a; b; k 2 k and u ; v 2 u(k). we show for any idempotent e 2 k, ( a 0 b 0 ) is e-clean i ( a 0 u(vb + ka) 0 ) is e-clean and if ( a 0 b 0 ) is 0-clean, ( ua 0 u(vb + ka) 0 ) is too.

the edge tenacity te(g) of a graph g is dened as:te(g) = min {[|x|+τ(g-x)]/[ω(g-x)-1]|x ⊆ e(g) and ω(g-x) > 1} where the minimum is taken over every edge-cutset x that separates g into ω(g - x) components, and by τ(g - x) we denote the order of a largest component of g. the objective of this paper is to determine this quantity for split graphs. let g = (z; i; e) be a noncomplete connected spli...

‎Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge‎ ‎set E(G)‎. ‎The (first) edge-hyper Wiener index of the graph G is defined as‎: ‎$$WW_{e}(G)=sum_{{f,g}subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=frac{1}{2}sum_{fin E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$‎ ‎where de(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and de(f|G)=∑g€(G)de(f,g|G). ‎In thi...

in a graph g = (v,e), a subset s  v is said to be an open packing set if no two vertices of s have a common neighbour in g. the maximum cardinality of an open packing set is called the open packing number and is denoted by rho^o. this paper further studies on this parameter by obtaining some new bounds.