Mark Sequences In Digraphs

In Chapter 1, we present a brief introduction of digraphs and some def- initions. Chapter 2 is a review of scores in tournaments and oriented graphs. Also we have obtained several new results on oriented graph scores and we have given a new proof of Avery's theorem on oriented graph scores. In chap- ter 3, we have introduced the concept of marks in multidigraphs, non-negative integers attached to the vertices of multidigraphs. We have obtained several necessary and su cient conditions for sequences of non-negative integers to be mark sequences of some r-digraphs. We have derived stronger inequalities for these marks. Further we have characterized uniquely mark sequences in r-digraphs. This concept of marks has been extended to bipartite multidi- graphs and multipartite multidigraphs in chapter 4. There we have obtained characterizations for mark sequences in these types of multidigraphs and we have given algorithms for constructing corresponding multidigraphs. Chap- ter 5 deals with imbalances and imbalance sequences in digraphs. We have generalized the concept of imbalances to oriented bipartite graphs and have obtained criteria for a pair of integers to be the pair of imbalance sequences of some oriented bipartite graph. We have shown the existence of an oriented bipartite graph whose imbalance set is the given set of integers

On graph energy, maximum degree and vertex cover number

For a simple graph $G$ with $n$ vertices and $m$ edges having adjacency eigenvalues $\lambda_1,\lambda_2, \dots,\lambda_n$, the energy $E(G)$ of $G$ is defined as $E(G)=\sum_{i=1}^{n} |\lambda_i|$. We obtain the upper bounds for $E(G)$ in terms of the vertex covering number $\tau$, the number of edges $m$, maximum vertex degree $d_1$ and second maximum vertex degree $d_2$ of the connected graph $G$. These upper bounds improve some recently known upper bounds for $E(G)$. Further, these upper bounds for $E(G)$ imply a natural extension to other energies like distance energy and Randi\'{c} energy associated to a connected graph $G$

On imbalances in multipartite multidigraphs

A k-partite r-digraph(multipartite multidigraph) (or briefly MMD)(k ≥ 3, r ≥ 1) is the result of assigning a direction to each edge of a k-partite multigraph that is without loops and contains at most r edges between any pair of vertices from distinct parts. Let D(X1, X2, ⋯, Xk) be a k-partite r-digraph with parts Xi = {xi1, xi2, ⋯, xini}, 1 ≤ i ≤ k. Let dxij +  and dxij −  be respectively the outdegree and indegree of a vertex xij in Xi. Define axij (or simply aij) as aij = dxij +  − dxij −  as the imbalance of the vertex xij, 1 ≤ j ≤ ni. In this paper, we characterize the imbalances of k-partite r-digraphs and give a constructive and existence criteria for sequences of integers to be the imbalances of some k-partite r-digraph. Also, we show the existence of a k-partite r-digraph with the given imbalance set.</p