In previous literature Coykendall & Maney, as well as Axtell & Stickles, have discussed the concept of irreducible divisor graphs of elements in domains and ring with zero-divisors respectively, with two different definitions. In this paper we seek to look at the irreducible divisor graphs of ring elements under a hybrid definition of the two previous ones—in hopes that this graph will reveal s...

In a recent paper [9], the authors introduced the extended Q-bipartite double of an almost dual bipartite cometric association scheme. Since the association schemes arising from linked systems of symmetric designs are almost dual bipartite, this gives rise to a new infinite family of 4-class cometric schemes which are both Q-bipartite and Q-antipodal. These schemes, the schemes arising from lin...

Let G = (V ,E) be a simple connected graph and λ1(G) be the largest Laplacian eigenvalue of G. In this paper, we prove that: 1. λ1(G) = max{du +mu : u ∈ V } if and only if G is a regular bipartite or a semiregular bipartite graph, where du and mu denote the degree of u and the average of the degrees of the vertices adjacent to u, respectively. 2. λ1(G) = 2 + √ (r − 2)(s − 2) if and only if G is...

You’ve probably seen some polynomial-time algorithms for the problem of computing a maximum-weight matching of a bipartite graph. Many of these, like the Kuhn-Tucker algorithm [?] are “combinatorial algorithms,” meaning that all of its steps work directly with the graph. Linear programming is also an effective tool for solving many discrete optimization problems. For example, consider the follo...

You’ve probably seen some polynomial-time algorithms for the problem of computing a maximum-weight matching of a bipartite graph. Many of these, like the Kuhn-Tucker algorithm [9], are “combinatorial algorithms” that operate directly on the graph. Linear programming is also an effective tool for solving many discrete optimization problems. For example, consider the following linear programming ...

It is well-known that any graph has all real eigenvalues and a graph is bipartite if and only if its spectrum is symmetric with respect to the origin. We are interested in finding whether the permanental roots of a bipartite graph G have symmetric property as the spectrum of G. In this note, we show that the permanental roots of bipartite graphs are symmetric with respect to the real and imagin...

The minimum crossing number problem is among the oldest and most fundamental problems arising in the area of automatic graph drawing. In this paper, eight population-based meta-heuristic algorithms are utilized to tackle the minimum crossing number problem for two special types of graphs, namely complete graphs and complete bipartite graphs. A 2-page book drawing representation is employed for ...

Subgraph enumeration problems ask to output all subgraphs of an input graph that belongs to the specified graph class or satisfy the given constraint. These problems have been widely studied in theoretical computer science. As far, many efficient enumeration algorithms for the fundamental substructures such as spanning trees, cycles, and paths, have been developed. This paper addresses the enum...

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its ve...

A bipartite graph G = (U, V, E) is a chain graph [9] if there is a bijection π : {1, . . . , |U |} → U such that Γ (π (1)) ⊇ Γ (π (2)) ⊇ . . . ⊇ Γ (π (|U |)), where Γ is a function that maps a node to its neighbors. We give approximation algorithms for two variants of the Minimum Chain Completion problem, where we are given a bipartite graph G(U, V, E), and the goal is find the minimum set of e...