Signed degree sets in signed graphs

The set D of distinct signed degrees of the vertices in a signed graph G is called its signed degree set. In this paper, we prove that every non-empty set of positive (negative) integers is the signed degree set of some connected signed graph and determine the smallest possible order for such a signed graph. We also prove that every non-empty set of integers is the signed degree set of some connected signed graph

Imbalances in directed multigraphs

In a directed multigraph, the imbalance of a vertex $v_{i}$ is defined as $b_{v_{i}}=d_{v_{i}}^{+}-d_{v_{i}}^{-}$, where $d_{v_{i}}^{+}$ and $d_{v_{i}}^{-}$ denote the outdegree and indegree respectively of $v_{i}$. We characterize imbalances in directed multigraphs and obtain lower and upper bounds on imbalances in such digraphs. Also, we show the existence of a directed multigraph with a given imbalance set

The transfer and fate of Pb from sewage sludge amended soil in a multi-trophic food chain: a comparison with the labile elements Cd and Zn

The contamination of agroecosystems due to the presence of trace elements in commonly used agricultural materials is a serious issue. The most contaminated material is usually sewage sludge, and the sustainable use of this material within agriculture is a major concern. This study addresses a key issue in this respect, the fate of trace metals applied to soil in food chains. The work particularly addresses the transfer of Pb, which is an understudied element in this respect, and compares the transfer of Pb with two of the most labile metals, Cd and Zn. The transfer of these elements was determined from sludge-amended soils in a food chain consisting of Indian mustard (Brassica juncea), the mustard aphid (Lipaphis erysimi) and a predatory beetle (Coccinella septempunctata). The soil was amended with sludge at rates of 0, 5, 10 and 20 % (w/w). Results showed that Cd was readily transferred through the food chain until the predator trophic level. Zn was the most readily transferred element in the lower trophic levels, but transfer to aphids was effectively restricted by the plant regulating shoot concentration. Pb had the lowest level of transfer from soil to shoot and exhibited particular retention in the roots. Nevertheless, Pb concentrations were significantly increased by sludge amendment in aphids, and Pb was increasingly transferred to ladybirds as levels increased. The potential for Pb to cause secondary toxicity to organisms in higher trophic levels may have therefore been underestimated

On scores in tournaments

A tournament is an orientation of a complete simple graph. The score of a vertex in a tournament is the outdegree of the vertex. In this paper, we obtain various results on the scores in tournaments

Tournaments, oriented graphs and football sequences

Consider the result of a soccer league competition where n teams play each other exactly once. A team gets three points for each win and one point for each draw. The total score obtained by each team vi is called the f-score of vi and is denoted by fi. The sequences of all f-scores [fi]i=1n$\left[ {{\rm{f}}_{\rm{i}} } \right]_{{\rm{i}} = 1}^{\rm{n}}$ arranged in non-decreasing order is called the f-score sequence of the competition. We raise the following problem: Which sequences of non-negative integers in non-decreasing order is a football sequence, that is the outcome of a soccer league competition. We model such a competition by an oriented graph with teams represented by vertices in which the teams play each other once, with an arc from team u to team v if and only if u defeats v. We obtain some necessary conditions for football sequences and some characterizations under restrictions

A NOTE ON SIGNED DEGREE SETS IN SIGNED BIPARTITE GRAPHS

A signed bipartite graph G(U, V) is a bipartite graph in which each edge is assigned a positive or a negative sign. The signed degree of a vertex x in G(U, V) is the number of positive edges incident with x less the number of negative edges incident with x. The set S of distinct signed degrees of the vertices of G(U, V) is called its signed degree set. In this paper, we prove that every set of integers is the signed degree set of some connected signed bipartite graph. 1