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Metric graph theory and geometry: a survey
 CONTEMPORARY MATHEMATICS
"... The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of general ..."
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Cited by 44 (14 self)
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The article surveys structural characterizations of several graph classes defined by distance properties, which have in part a general algebraic flavor and can be interpreted as subdirect decomposition. The graphs we feature in the first place are the median graphs and their various kinds of generalizations, e.g., weakly modular graphs, or fibercomplemented graphs, or l1graphs. Several kinds of l1graphs admit natural geometric realizations as polyhedral complexes. Particular instances of these graphs also occur in other geometric contexts, for example, as dual polar graphs, basis graphs of (even ∆)matroids, tope graphs, lopsided sets, or plane graphs with vertex degrees and face sizes bounded from below. Several other classes of graphs, e.g., Helly graphs (as injective objects), or bridged graphs (generalizing chordal graphs), or treelike graphs such as distancehereditary graphs occur in the investigation of graphs satisfying some basic properties of the distance function, such as the Helly property for balls, or the convexity of balls or of the neighborhoods of convex sets, etc. Operators between graphs or complexes relate some of the
Strongly distancebalanced graphs and graph products
"... A graph G is strongly distancebalanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v)−1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distancebalanced if and only if both ..."
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Cited by 3 (1 self)
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A graph G is strongly distancebalanced if for every edge uv of G and every i ≥ 0 the number of vertices x with d(x, u) = d(x, v)−1 = i equals the number of vertices y with d(y, v) = d(y, u) − 1 = i. It is proved that the strong product of graphs is strongly distancebalanced if and only if both factors are strongly distancebalanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distancebalanced if and only if both factors are strongly distancebalanced. Additionally, a new characterization of distancebalanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given.
Wiener Dimension: Fundamental Properties and (5,0)Nanotubical Fullerenes
, 2014
"... The Wiener dimension of a connected graph is introduced as the number of different distances of its vertices. For any integer D and any integer k, a graph of diameter D and of Wiener dimension k is constructed. An infinite family of nonvertextransitive graphs with Wiener dimension 1 is presented ..."
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The Wiener dimension of a connected graph is introduced as the number of different distances of its vertices. For any integer D and any integer k, a graph of diameter D and of Wiener dimension k is constructed. An infinite family of nonvertextransitive graphs with Wiener dimension 1 is presented and it is proved that a graph of dimension 1 is 2connected. It is shown that the (5, 0)nanotubical fullerene graph on 10k (k ≥ 3) vertices has Wiener dimension k. As a consequence the Wiener index of these fullerenes is obtained.
On the remoteness function in median graphs
"... A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hyperc ..."
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A profile on a graph G is any nonempty multiset whose elements are vertices from G. The corresponding remoteness function associates to each vertex x ∈ V (G) the sum of distances from x to the vertices in the profile. Starting from some nice and useful properties of the remoteness function in hypercubes, the remoteness function is studied in arbitrary median graphs with respect to their isometric embeddings in hypercubes. In particular, a relation between the vertices in a median graph G whose remoteness function is maximum (antimedian set of G) with the antimedian set of the host hypercube is found. While for odd profiles the antimedian set is an independent set that lies in the strict boundary of a median graph, there exist median graphs in which special even profiles yield a constant remoteness function. We characterize such median graphs in two ways: as the graphs whose periphery transversal number is 2, and as the graphs with the geodetic number equal to 2. Finally, we present an algorithm that, given a graph G on n vertices and m edges, decides in O(m log n) time whether G is a median graph with geodetic number 2.
doi:10.5402/2012/583671 Research Article Median Sets and Median Number of a Graph
"... the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances t ..."
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the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A profile is a finite sequence of vertices of a graph. The set of all vertices of the graph which minimises the sum of the distances to the vertices of the profile is the median of the profile. Any subset of the vertex set such that it is the median of some profile is called a median set. The number of median sets of a graph is defined to be the median number of the graph. In this paper, we identify the median sets of various classes of graphs such asKp − e,Kp,q for P> 2, and wheel graph and so forth. The median numbers of these graphs and hypercubes are found out, and an upper bound for the median number of even cycles is established. We also express the median number of a product graph in terms of the median number of their factors. 1.