A (d − 1)-dimensional simplicial complex is called balanced if its underlying graph admits a proper d-coloring. We show that many well-known face enumeration results have natural balanced analogs (or at least conjectural analogs). Specifically, we prove the balanced analog of the celebrated Lower Bound Theorem for normal pseudomanifolds and characterize the case of equality; we introduce and ch...

For a graph G = (V,E) of even order, a partition (V1, V2) of the vertices is said to be perfectly balanced if |V1| = |V2| and the numbers of edges in the subgraphs induced by V1 and V2 are equal. For a base graph H define a random graph G(H, p) by turning every non-edge of H into an edge and every edge of H into a nonedge independently with probability p. We show that for any constant ǫ there i...

The diameter of a connected graph $G$, denoted by $diam(G)$, is the maximum distance between any pair of vertices of $G$. Let $L(G)$ be the line graph of $G$. We establish necessary and sufficient conditions under which for a given integer $k geq 2$, $diam(L(G)) leq k$.

the wiener polarity index wp(g) of a molecular graph g of order n is the number ofunordered pairs of vertices u, v of g such that the distance d(u,v) between u and v is 3. in anearlier paper, some extremal properties of this graph invariant in the class of catacondensedhexagonal systems and fullerene graphs were investigated. in this paper, some new bounds forthis graph invariant are presented....

A graph curve is a nodal curve C such that each irreducible component of C is smooth, rational and meets exactly 3 others (Bayer and Eisenbud). Here we classify the graph curves with gonality 1 with respect to balanced line bundles in the sense of L. Caporaso.

The quantity g (k) (v) was introduced in [6] as the minimum number of blocks necessary in a pairwise balanced design on v elements, subject to the condition that the longest block has length k. Recently, we have needed to use all possibilities for such minimal covering designs, and we record all non-isomorphic solutions to the problem for v ≤ 13.

‎A set of vertices $S$ of a graph $G$ is called a fixing set of $G$‎, ‎if only the trivial automorphism of $G$ fixes every vertex in $S$‎. ‎The fixing number of a graph is the smallest cardinality of a fixing‎ ‎set‎. ‎The fixed number of a graph $G$ is the minimum $k$‎, ‎such that ‎every $k$-set of vertices of $G$ is a fixing set of $G$‎. ‎A graph $G$‎ ‎is called a $k$-fixed graph‎, ‎if its fix...

Balanced structure and parameter search is critical to evolutionary design with genetic programming (GP). Structure Fitness Sharing (SFS), based on a structure labeling technique, is proposed to maintain the structural diversity of a population and combat premature convergence of structures. SFS achieves balanced structure and parameter search by applying fitness sharing to each unique structur...