Let G = (V, E) be a simple graph. A vertex labeling f : V → {0, 1} induces a partial edge labeling f ∗ : E → {0, 1} defined by f ∗(uv) = f(u) 84 KWONG, LEE, LO, SU AND WANG if and only if f(u) = f(v). For i = 0, 1, let vf(i) = |{v ∈ V : f(v) = i}|, and ef (i) = |{e ∈ E : f ∗(e) = i}|. A graph G is uniformly balanced if |ef (0)−ef (1)| ≤ 1 for any vertex labeling f that satisfies |vf(0)−vf (1)| ...

In [Structural properties and decomposition of linear balanced matrices, Mathematical Programming, 55:129–168, 1992], Conforti and Rao conjectured that every balanced bipartite graph contains an edge that is not the unique chord of a cycle. We prove this conjecture for balanced bipartite graphs that do not contain a cycle of length 4 (also known as linear balanced bipartite graphs), and for bal...

Distance-balanced graphs are introduced as graphs in which every edge uv has the following property: the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. Basic properties of these graphs are obtained. The new concept is connected with symmetry conditions in graphs and local operations on graphs are studied with respect to it. Distance-balanced C...

It is shown that the graphs for which the Szeged index equals ‖G‖·|G| 2 4 are precisely connected, bipartite, distance-balanced graphs. This enables to disprove a conjecture proposed in [Some new results on distance-based graph invariants, European J. Combin. 30 (2009) 1149–1163]. Infinite families of counterexamples are based on the Handa graph, the Folkman graph, and the Cartesian product of ...

We generalize chain enumeration in graded partially ordered sets by relaxing the graded, poset and Eulerian requirements. The resulting balanced digraphs, which include the classical Eulerian posets having an R-labeling, implies the existence of the (non-homogeneous) cd-index, a key invariant for studying inequalities for the flag vector of polytopes. Mirroring Alexander duality for Eulerian po...

A group-labeled graph is a graph whose vertices and edges have been assigned labels from some abelian group. The weight of a subgraph of a group-labeled graph is the sum of the labels of the vertices and edges in the subgraph. A group-labeled graph is said to be balanced if the weight of every cycle in the graph is zero. We give a characterization of balanced group-labeled graphs that generaliz...

A graph X is said to be distance–balanced if for any edge uv of X , the number of vertices closer to u than to v is equal to the number of vertices closer to v than to u. A graph X is said to be strongly distance–balanced if for any edge uv of X and any integer k, the number of vertices at distance k from u and at distance k+1 from v is equal to the number of vertices at distance k + 1 from u a...

The concept of strongly balanced graph is introduced. It is shown that there exists a strongly balanced graph with u vertices and e edges if and only if I s u 1 s e s (2"). This result is applied to a classic question of Erdos and Renyi: What is the probability that a random graph on n vertices contains a given graph? A rooted version of this problem is also solved.