Distance in graphs

The structure of the centroid in a Ptolemaic graph

For a connected graph G of order p ≥ 2 and a set W ⊆ V(G), a tree T contained in G is a Steiner tree with respect to W if T is a tree of minimum order with W ⊆ V(T). The set S(W) consists of all vertices in G that lie on some Steiner tree with respect to W. The set W is a Steiner set for G if S(W) = V(G). The Steiner number s(G) of G is the minimum cardinality of its Steiner sets and any Steiner set of cardinality s(G) is a minimum Steiner set of G. A geodetic set of G is a set S of vertices such that every vertex of G is contained in a geodesic joining some pair of vertices of S. The geodetic number g(G) of G is the minimum cardinality of its geodetic sets and any geodetic set of cardinality g(G) is a minimum geodetic set of G. A vertex v is an extreme vertex of a graph G if the subgraph induced by its neighbors is complete. The number of extreme vertices in G is its extreme order ex(G). A graph G is an extreme Steiner graph if s(G) = ex(G), and an extreme geodesic graph if g(G) = ex(G). Extreme Steiner graphs of order p with Steiner number p - 1 are characterized. It is shown that every pair a, b of integers with 0 ≤ a ≤ b is realizable as the extreme order and Steiner number, respectively, of some graph. For positive integers r, d and l ≥ 2 with r < d ≤ 2r, it is shown that there exists an extreme Steiner graph G of radius r, diameter d, and Steiner number l. For integers p, d and k with 2 ≤ d < p, 2 ≤ k < p and p - d - k + 2 ≥ 0, there exists an extreme Steiner graph G of order p, diameter d and Steiner number k. It is shown that for every pair a, b of integers with 3 ≤ a < b and b = a + 1, there exists an extreme Steiner graph G with s(G) = a and g(G) = b that is not an extreme geodesic graph. It is shown that for every pair a, b of integers with 3 ≤ a < b, there exists an extreme geodesic graph G with g(G) = a and s(G) = b that is not an extreme Steiner graph.

background

Extreme steiner graphs

Let G be a connected graph. A spanning tree T of G is a tree t-spanner if the distance between any two vertices in T is at most t times their distance in G. If their distances in T and G differ by at most t, then T is an additive tree t-spanner of G. In this paper, we show that any permutation graph has an additive tree 2-spanner, and it can be found in O(n) time sequentially or in O(log n) time with O(n/log n) processors on the EREW PRAM computational model by using a previously published algorithm for finding a tree 3-spanner of a permutation graph.

Additive tree 2-spanners of permutation graphs

The circular distance energy of a simple connected graph \(G\) is defined as the sum of the absolute values of its eigen values of the circular distance matrix of \(G\). In this paper, the bounds for circular distance energy is obtained. Also the circular distance energy and the circular distance laplacian energy of certain graphs via circular distance energy are derived.

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Distance in graphs

Citations

The structure of the centroid in a Ptolemaic graph

Extreme steiner graphs

Additive tree 2-spanners of permutation graphs

A Note on Circular Distance Energy and Circular Distance Laplacian Energy

some graph theoretic invariants

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