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1
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Radii and centers in iterated line digraphs

100%
Knor M., Niepel L. u.
Discussiones Mathematicae Graph Theory
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1996
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tom 16
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nr 1
17-26
EN
We show that the out-radius and the radius grow linearly, or "almost" linearly, in iterated line digraphs. Further, iterated line digraphs with a prescribed out-center, or a center, are constructed. It is shown that not every line digraph is admissible as an out-center of line digraph.
2
Content available remote

Characterization Of Super-Radial Graphs

100%
Kathiresan K. M., Marimuthu G., Parameswaran C.
Discussiones Mathematicae Graph Theory
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2014
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tom 34
|
nr 4
829-848
EN
In a graph G, the distance d(u, v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius, r(G), of the graph and the maximum eccentricity is called the diameter, d(G), of the graph. The super-radial graph R*(G) based on G has the vertex set as in G and two vertices u and v are adjacent in R*(G) if the distance between them in G is greater than or equal to d(G) − r(G) + 1 in G. If G is disconnected, then two vertices are adjacent in R*(G) if they belong to different components. A graph G is said to be a super-radial graph if it is a super-radial graph R*(H) of some graph H. The main objective of this paper is to solve the graph equation R*(H) = G for a given graph G.
3
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Further results on radial graphs

100%
Kathiresan K., Marimuthu G.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 1
75-83
EN
In a graph G, the distance d(u,v) between a pair of vertices u and v is the length of a shortest path joining them. The eccentricity e(u) of a vertex u is the distance to a vertex farthest from u. The minimum eccentricity is called the radius of the graph and the maximum eccentricity is called the diameter of the graph. The radial graph R(G) based on G has the vertex set as in G, two vertices u and v are adjacent in R(G) if the distance between them in G is equal to the radius of G. If G is disconnected, then two vertices are adjacent in R(G) if they belong to different components. The main objective of this paper is to characterize graphs G with specified radius for its radial graph.
4
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Lower bounds for the domination number

100%
Delaviña E., Pepper R., Waller B.
Discussiones Mathematicae Graph Theory
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2010
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tom 30
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nr 3
475-487
EN
In this note, we prove several lower bounds on the domination number of simple connected graphs. Among these are the following: the domination number is at least two-thirds of the radius of the graph, three times the domination number is at least two more than the number of cut-vertices in the graph, and the domination number of a tree is at least as large as the minimum order of a maximal matching.
5
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Triangle-free planar graphs with minimum degree 3 have radius at least 3

88%
Kim S.-J., West D.
Discussiones Mathematicae Graph Theory
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2008
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tom 28
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nr 3
563-566
EN
We prove that every triangle-free planar graph with minimum degree 3 has radius at least 3; equivalently, no vertex neighborhood is a dominating set.
6
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Notes on the independence number in the Cartesian product of graphs

88%
Abay-Asmerom G., Hammack R., Larson C., Taylor D.
Discussiones Mathematicae Graph Theory
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2011
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tom 31
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nr 1
25-35
EN
Every connected graph G with radius r(G) and independence number α(G) obeys α(G) ≥ r(G). Recently the graphs for which equality holds have been classified. Here we investigate the members of this class that are Cartesian products. We show that for non-trivial graphs G and H, α(G ☐ H) = r(G ☐ H) if and only if one factor is a complete graph on two vertices, and the other is a nontrivial complete graph. We also prove a new (polynomial computable) lower bound α(G ☐ H) ≥ 2r(G)r(H) for the independence number and we classify graphs for which equality holds. The second part of the paper concerns independence irreducibility. It is known that every graph G decomposes into a König-Egervary subgraph (where the independence number and the matching number sum to the number of vertices) and an independence irreducible subgraph (where every non-empty independent set I has more than |I| neighbors). We examine how this decomposition relates to the Cartesian product. In particular, we show that if one of G or H is independence irreducible, then G ☐ H is independence irreducible.
7
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Eccentric distance sum index for some classes of connected graphs

75%
Bielak H., Broniszewska K.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2017
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tom 71
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nr 2
EN
In this paper we show some properties of the eccentric distance sum index which is defined as follows \(\xi^{d}(G)=\sum_{v \in V(G)}D(v) \varepsilon(v)\). This index is widely used by chemists and biologists in their researches. We present a lower bound of this index for a new class of graphs.
8
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Minimal non-selfcentric radially-maximal graphs of radius 4

75%
Knor M.
Discussiones Mathematicae Graph Theory
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2007
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tom 27
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nr 3
603-610
EN
There is a hypothesis that a non-selfcentric radially-maximal graph of radius r has at least 3r-1 vertices. Using some recent results we prove this hypothesis for r = 4.
9
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p-Wiener intervals and p-Wiener free intervals

75%
Kathiresan K., Arockiaraj S.
Discussiones Mathematicae Graph Theory
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2012
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tom 32
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nr 1
121-127
EN
A positive integer n is said to be Wiener graphical, if there exists a graph G with Wiener index n. In this paper, we prove that any positive integer n(≠ 2,5) is Wiener graphical. For any positive integer p, an interval [a,b] is said to be a p-Wiener interval if for each positive integer n ∈ [a,b] there exists a graph G on p vertices such that W(G) = n. For any positive integer p, an interval [a,b] is said to be p-Wiener free interval (p-hyper-Wiener free interval) if there exist no graph G on p vertices with a ≤ W(G) ≤ b (a ≤ WW(G) ≤ b). In this paper, we determine some p-Wiener intervals and p-Wiener free intervals for some fixed positive integer p.
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