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On the crossing numbers of G □ Cₙ for graphs G on six vertices

Emília Draženská, Marián Klešč (2011)

Discussiones Mathematicae Graph Theory

The crossing numbers of Cartesian products of paths, cycles or stars with all graphs of order at most four are known. The crossing numbers of G☐Cₙ for some graphs G on five and six vertices and the cycle Cₙ are also given. In this paper, we extend these results by determining crossing numbers of Cartesian products G☐Cₙ for some connected graphs G of order six with six and seven edges. In addition, we collect known results concerning crossing numbers of G☐Cₙ for graphs G on six vertices.

On the Decompositions of Complete Graphs into Cycles and Stars on the Same Number of Edges

Atif A. Abueida, Chester Lian (2014)

Discussiones Mathematicae Graph Theory

Let Cm and Sm denote a cycle and a star on m edges, respectively. We investigate the decomposition of the complete graphs, Kn, into cycles and stars on the same number of edges. We give an algorithm that determines values of n, for a given value of m, where Kn is {Cm, Sm}-decomposable. We show that the obvious necessary condition is sufficient for such decompositions to exist for different values of m.

On The Determinant of q-Distance Matrix of a Graph

Hong-Hai Li, Li Su, Jing Zhang (2014)

Discussiones Mathematicae Graph Theory

In this note, we show how the determinant of the q-distance matrix Dq(T) of a weighted directed graph G can be expressed in terms of the corresponding determinants for the blocks of G, and thus generalize the results obtained by Graham et al. [R.L. Graham, A.J. Hoffman and H. Hosoya, On the distance matrix of a directed graph, J. Graph Theory 1 (1977) 85-88]. Further, by means of the result, we determine the determinant of the q-distance matrix of the graph obtained from a connected weighted graph...

On the diameter of the intersection graph of a finite simple group

Xuanlong Ma (2016)

Czechoslovak Mathematical Journal

Let G be a finite group. The intersection graph Δ G of G is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper nontrivial subgroups of G , and two distinct vertices X and Y are adjacent if X ∩ Y ≠ 1 , where 1 denotes the trivial subgroup of order 1 . A question was posed by Shen (2010) whether the diameters of intersection graphs of finite non-abelian simple groups have an upper bound. We answer the question and show that the diameters of intersection...

On the distance function of a connected graph

Ladislav Nebeský (2008)

Czechoslovak Mathematical Journal

An axiomatic characterization of the distance function of a connected graph is given in this note. The triangle inequality is not contained in this characterization.

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