We investigate relations between different width parameters of graphs, in particular balanced separator number, treewidth, and cycle rank. Our main result states that a graph with balanced separator number k has treewidth at least k but cycle rank at most k · ( 1 + log nk ) , thus refining the previously known bounds, as stated by Robertson and Seymour (1986) and by Bodlaender et al. (1995). Fu...

In this article, we investigate the controllability of multi-agent systems with leaders as control inputs, where the interconnection is directed and weighted. We employ weight-balanced partition to classify the interconnection graphs, and study the controllable subspaces with given nontrivial weight-balanced partition. We also provide two necessary and sufficient graph conditions on structural ...

A graph G of order n is called t-edge-balanced if G satisfies the property that there exists a positive λ for which every graph of order n and size t is contained in exactly λ distinct subgraphs of Kn isomorphic to G. We call λ the index of G. In this article, we obtain new infinite families of 2-edge-balanced graphs. C ⃝ 2013 Wiley Periodicals, Inc. J. Combin. Designs 22: 291–305, 2014

A problem is said to be GI-complete if it is provably as hard as graph isomorphism; that is, there is a polynomial-time Turing reduction from the graph isomorphism problem. It is known that the GI problem is GI-complete for some special graph classes including regular graphs, bipartite graphs, chordal graphs and split graphs. In this paper, we prove that deciding isomorphism of double split gra...

A graph G is said to be distance-balanced if for any edge uv of G, the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Let GP (n, k) be a generalized Petersen graph. Jerebic, Klavžar, and Rall [Distance-balanced graphs, Ann. Comb. 12 (2008) 71–79] conjectured that: For any integer k > 2, there exists a positive integer n0 such that the GP (n, k...

A model of taxation for cooperative n-person games is introduced where proper coalitions are taxed proportionally to their value. Games with non-empty core under taxation at rate e are a-balanced. Sharp bounds on e in matching games on (not necessarily bipartite) graphs are established. Upper and lower bounds on the smallest e in bin packing games are derived and euclidean random TSP games are ...