introduction: brachytherapy is a type of radiotherapy in which radioactive sources are used in proximity of tumors normally for treatment of malignancies in the head, prostate and cervix. materials and methods: the cs-137 selectron source is a low-dose-rate (ldr) brachytherapy source used in a remote afterloading system for treatment of different cancers. this system uses active and inactive sp...

A Steiner tree for a set S of vertices in a connected graph G is a connected subgraph of G with a smallest number of edges that contains S. The Steiner interval I (S) of S is the union of all the vertices of G that belong to some Steiner tree for S. If S = {u, v}, then I (S) = I [u, v] is called the interval between u and v and consists of all vertices that lie on some shortest u–v path in G. T...

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. The degree resistance distance of a graph G is defined as D R (G) =  {u,v}⊆V (G) [d(u) + d(v)]R(u, v), where d(u) is the degree of the vertex u, and R(u, v) the resistance distance between the vertices u and v. Let Cact(n; t) be the set of all cacti possessing n vertices and t cy...

Let G be a simple connected graph with vertex set V = {1, 2, . . . , n}. Let d(i, j) denote the distance between vertices i and j. For i ∈ V , the degree of i and the average of the degree of the vertices adjacent to i are denoted by di and mi, respectively. The 2-degree of vertex i is denoted by ti, which is the sum of degrees of the vertices adjacent to i, that is ti = midi. Let Ni be the sum...

CAT(0) metric spaces constitute a far-reaching common generalization of Euclidean and hyperbolic spaces and simple polygons: any two points x and y of a CAT(0) metric space are connected by a unique shortest path γ(x, y). In this paper, we present an efficient algorithm for answering two-point distance queries in CAT(0) rectangular complexes and two of theirs subclasses, ramified rectilinear po...

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...

In this paper, we are concerned with finite undirected graphs without loops or multiple edges. The distance between two vertices u and v of a graph G, denoted by dG(u, v) or d(u, v) for short, is the length of a shortest (u, v)-path in G. Let d(G) denote the maximum distance over all pairs of vertices of graph G, namely its diameter of G. If denote by d(G, k) the number of pairs of vertices of ...

Throughout this paper, we assume Γ is a connected finite undirected graph without loops or multiple edges. We identify Γ with the set of vertices. For vertices u and x in Γ, let ∂(u, x) denote the distance between u and x in Γ, i.e., the length of a shortest path connecting u and x. Let d = d(Γ) denote the diameter of Γ, i.e., the maximal distance between any two vertices in Γ. Let Γi(u) = { y ...

Abstract We completely characterize when the free effective resistance of an infinite graph whose vertices have finite degrees can be expressed in terms simple hitting probabilities random walk on graph.