For two graph H and G, the Ramsey number r(H,G) is the smallest positive integer n such that every red-blue edge coloring of the complete graph Kn on n vertices contains either a red copy of H or a blue copy of G. Motivated by questions of Erdős and Harary, in this note we study how the Ramsey number r(Ks, G) depends on the size of the graph G. For s ≥ 3, we prove that for every G with m edges,...

Given a simple connected graph G = (V, E) the geodetic closure I[S] ⊂ V of a subset S of V is the union of all sets of nodes lying on some geodesic (or shortest path) joining a pair of nodes vk, vl ∈ S. The geodetic number, denoted by g(G), is the smallest cardinality of a node set S∗ such that I[S∗] = V . In “The geodetic number of a graph”, Mathematical and Computer Modelling 17 (June 1993) 8...

A topological index plays an important role in characterising various physical properties, biological activities, and chemical reactivities of a molecular graph. The Hosoya polynomial is used to evaluate the distance-based indices such as Wiener index, hyper-Wiener Harary Tratch – Stankevitch Zefirov index. In present study, we determine closed form for third type chain hex-derived network dime...

A graph is said to be (k, l)-connected if the resulted graph after removing of any k vertices and (l−1) edges or removing of any (k−1) vertices and l edges is still connected. Beineke and Harary (1967) (see [1]) claimed to prove that there should be k+ l edge-disjoint paths, of which k are vertex-disjoint, between any pair of vertices if the graph has (k, l)-connectivity. However, Mader (1979) ...

Jochen Reibenwein, Dietmar Pils, Peter Horak, Birgit Tomicek, Gregor Goldner, Nina Worel, Katarzyna Elandt, and Michael Krainer* DivisionofOncology,Departmentof InternalMedicine I,Medical Universityof Vienna,Vienna, Austria Departmentof RadiotherapyandRadiobiology,Medical Universityof Vienna,Vienna, Austria Departmentof Bloodgroup SerologyandTransfusionMedicine,Medical Universityof Vienna,Vienn...

The graphs studied in this paper are finite, undirected, without loops and multiple edges. A graph is called unretractive if it is isomorphic to each of its retracts. Unretractivity of graphs has been studied as a tool for generation and investigation of transformation groups. In particular the automorphism groups of graph products have been studied by Harary, Sabidussi, Hemminger, and Imrich a...

The ith cycle minor of a permutation p of the set {1, 2, . . . , n} is the permutation formed by deleting an entry i from the decomposition of p into disjoint cycles and reducing each remaining entry larger than i by 1. In this paper, we show that any permutation of {1, 2, . . . , n} can be reconstructed from its set of cycle minors if and only if n ≥ 6. We then use this to provide an alternate...

A set S of vertices of a graph G= (V ,E) with no isolated vertex is a total dominating set if every vertex of V (G) is adjacent to some vertex in S. The total domination number t(G) is the minimum cardinality of a total dominating set ofG. The total domination subdivision number sd t (G) is the minimum number of edges that must be subdivided in order to increase the total domination number. We ...

A stable set in a graph G is a set of pairwise non-adjacent vertices and α(G) is the size of a maximum stable set in the graph G. The polynomial I(G; x) = s0 + s1x + s2x + ... + sαx, α = α(G), is called the independence polynomial of G (Gutman and Harary, 1983), where sk is the number of stable sets of cardinality k in G. I(G; x) is partial unimodal if the sequence of its coefficients (sk) is p...

An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of cardinality k in G, then I(G;x) = s0 + s1x+ s2x 2 + ...+ sαx , α = α (G) , is called the independence polynomial of G ...