Matrix symmetrization and several related problems have an extensive literature, with a recurring ambiguity regarding their complexity status and their relation to graph isomorphism. We present a short survey of these problems to clarify their status. In particular we recall results from the literature showing that matrix symmetrization is in fact NP-hard, while graph isomorphism is still an op...

In [1], Afkhami and Khashyarmanesh introduced the cozero-divisor graph of a ring, Γ′(R), which examines relationships between principal ideals. We continue investigating the algebraic implications of the graph by developing the reduced cozero-divisor graph, which is a simpler analog. Acknowledgements: This research was undertaken at the Wabash College Mathematics REU in Crawfordsville, Indiana....

There is a natural graph associated to the zero-divisors of a commutative semiring with non-zero identity. In this article we essentially study zero-divisor graphs with respect to primal and non-primal ideals of a commutative semiring R and investigate the interplay between the semiring-theoretic properties of R and the graph-theoretic properties of ΓI(R) for some ideal I of R. We also show tha...

We provide a process to extend any bipartite diametrical graph of diameter 4 to an S-graph of the same diameter and partite sets. For a bipartite diametrical graph of diameter 4 and partite sets U and W , where 2m |U| ≤ |W |, we prove that 2 is a sharp upper bound of |W | and construct an S-graph G 2m, 2 in which this upper bound is attained, this graph can be viewed as a generalization of the ...

The permanental polynomial of a graph G is π(G,x) , per(xI −A(G)). From the result that a bipartite graph G admits an orientation Ge such that every cycle is oddly oriented if and only if it contains no even subdivision of K2,3, Yan and Zhang showed that the permanental polynomial of such a bipartite graph G can be expressed as the characteristic polynomial of the skew adjacency matrix A(Ge). I...

The mirror (or bipartite complement) mir(B) of a bipartite graph B = (X,Y,E) has the same color classes X and Y as B, and two vertices x ∈ X and y ∈ Y are adjacent in mir(B) if and only if xy / ∈ E. A bipartite graph is chordal bipartite if none of its induced subgraphs is a chordless cycle with at least six vertices. In this paper, we deal with chordal bipartite graphs whose mirror is chordal ...

Lexicographic Breadth First Search (LBFS) is one of fundamental graph search algorithms that has numerous applications, including recognition of graph classes, computation of graph parameters, and detection of certain graph structures. The well-known result on the end-vertices of LBFS of chordal graphs has tempted researchers to study the end-vertices of LBFS of various classes of graphs, inclu...