Sandi Klavžar

The Szeged and the Wiener Index of Graphs

The Szeged index Sz is a recently introduced graph invariant, having applications in chemistry. In this paper, a formula for the Szeged index of Cartesian product graphs is obtained and some other composite graphs are considered. We also prove that for all connected graphs, Sz is greater than or equal to the sum of distances between all vertices. A conjecture concerning the maximum value of Sz is put forward.

Wiener number of vertex-weighted graphs and a chemical application

The Wiener number W(G) of a graph G is the sum of distances between all pairs of vertices of G. If (G, w) is a vertex-weighted graph, then the Wiener number W(G, w) of (G, w) is the sum, over all pairs of vertices, of products of weights of the vertices and their distance. For G being a partial binary Hamming graph, a formula is given for computing W(G, w) in terms of a binary Hamming labeling of G. This result is applied to prove that W(PH) = W(HS) + 36W(ID), where PH is a phenylene, HS a pertinently vertex-weighted hexagonal squeeze of PH, and ID the inner dual of the hexagonal squeeze.

Isomorphic components of Kronecker product of bipartite graphs

Weichsel (Proc. Amer. Math. Soc. 13 (1962), 47-52) proved that the Kronecker product of two connected bipartite graphs consists of two connected components. A condition on the factor graphs is presented which ensures that such components are isomorphic. It is demonstrated that several familiar and easily constructible graphs are amenable to that condition. A partial converse is proved for the above condition and it is conjectured that the converse is true in general. Math. Subj. Class. (1991): 05C60 Key terms: Kronecker product, bipartite graphs, graph isomorphism ∗This work was supported in part by the Ministry of Science and Technology of Slovenia under the grants P1-0206-101 and J1-7036.

On the complexity of recognizing Hamming graphs and related classes of graphs

Abstract This paper contains a new algorithm that recognizes whether a given graph G is a Hamming graph, i.e. a Cartesian product of complete graphs, in O(m) time and O(n 2 ) space. Here m and n denote the numbers of edges and vertices of G, respectively. Previously this was only possible in O(m log n) time. Moreover, we present a survey of other recognition algorithms for Hamming graphs, retracts of Hamming graphs and isometric subgraphs of Hamming graphs. Special emphasis is also given to the bipartite case in which these classes are reduced to binary Hamming graphs, median graphs and partial binary Hamming graphs.

Coloring graph products—a survey

Abstract There are four standard products of graphs: the direct product, the Cartesian product, the strong product and the lexicographic product. The chromatic number turned out to be an interesting parameter on all these products, except on the Cartesian one. A survey is given on the results concerning the chromatic number of the three relevant products. Some applications of product colorings are also included.

Recognizing median graphs in subquadratic time

Abstract Motivated by a dynamic location problem for graphs, Chung, Graham and Saks introduced a graph parameter called windex. Graphs of windex 2 turned out to be, in graph-theoretic language, retracts of hypercubes. These graphs are also known as median graphs and can be characterized as partial binary Hamming graphs satisfying a convexity condition. In this paper an O(n 3 2 log n) algorithm is presented to recognize these graphs. As a by-product we are also able to isometrically embed median graphs in hypercubes in O(m log n) time.

Distances in benzenoid systems: further developments

Abstract In this note we present some new results on distances in benzenoids. An algorithm is presented which, for a given benzenoid system G bounded by a simple circuit Z with n vertices, computes the Wiener index of G in O( n ) time. Also we show that benzenoid systems have a convenient dismantling scheme, which can be derived by applying breadth-first search to their dual graphs. Our last result deals with the clustering problem of sets of atoms of benzenoids systems. We show how the k -means clustering algorithm (for points in Euclidean space) can be efficiently implemented in the case of benzenoids.

Bounds for the Schultz Molecular Topological Index of Benzenoid Systems in Terms of the Wiener Index

Let MTI and W be the Schultz molecular topological index and the Wiener index, respectively, of a benzenoid system. It has been shown previously [Klavžar, S.; Gutman, I. J. Chem. Inf. Comput. Sci. 1996, 36, 1001−1003] that MTI is bounded as 4W < MTI < 6.93 W. We now improve this result by deducing the estimates 4W + λ1W2/3 + λ2W1/3 − 15 < MTI < 6W + λ3 W2/5 − λ4 W1/6 where λ1 = 4.400, λ2 = 1.049, λ3 = 14.760, and λ4 = 17.739.

An Euler-type formula for median graphs

Abstract Let G be a median graph on n vertices and m edges and let k be the number of equivalence classes of the Djokovic´s relation Θ defined on the edge-set of G . Then 2 n - m -k⩽2. Moreover, 2 n − m − k = 2 if and only if G is cube-free.

On a Vizing-like conjecture for direct product graphs

Abstract Let γ ( G ) be the domination number of a graph G , and let G × H be the direct product of graphs G and H . It is shown that for any k ⩾ 0 there exists a graph G such that γ ( G × G ) ⩽ γ ( G ) 2 − k . This in particular disproves a conjecture from [5].