Related papers: Minimal non-odd-transversal hypergraphs and minima…

It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to…

Let $G$ be a connected non-odd-bipartite hypergraph with even uniformity. The least H-eigenvalue of the signless Laplacian tensor of $G$ is simply called the least eigenvalue of $G$ and the corresponding H-eigenvectors are called the first…

A $k$-uniform hypergraph $G=(V,E)$ is called odd-bipartite ([5]), if $k$ is even and there exists some proper subset $V_1$ of $V$ such that each edge of $G$ contains odd number of vertices in $V_1$. Odd-bipartite hypergraphs are…

Bipartite graphs are a fundamental concept in graph theory with diverse applications. A graph is bipartite iff it contains no odd cycles, a characteristic that has many implications in diverse fields ranging from matching problems to the…

In order to investigate the non-odd-bipartiteness of even uniform hypergraphs, starting from a simple graph $G$, we construct a generalized power of $G$, denoted by $G^{k,s}$, which is obtained from $G$ by blowing up each vertex into a…

A graph is said to be edge-transitive if its automorphism group acts transitively on its edges. It is known that edge-transitive graphs are either vertex-transitive or bipartite. In this paper we present a complete classification of all…

A hypergraph is called uniform when every hyperedge contains the same number of vertices, otherwise, it is called non-uniform. In the real world, many systems give rise to non-uniform hypergraphs, such as email networks and co-authorship…

While the adjacency tensor of a bipartite 2-graph is a nonnegative biquadratic tensor, it is inherently reducible. To address this limitation, we introduce the concept of quasi-irreducibility in this paper. The adjacency tensor of a…

We characterize the bipartite graphs that minimize the (first-degree based) entropy, among all bipartite graphs of given size, or given size and (upper bound on the) order. The extremal graphs turn out to be complete bipartite graphs, or…

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite…

Motivated by the concept of well-covered graphs, we define a graph to be well-bicovered if every vertex-maximal bipartite subgraph has the same order (which we call the bipartite number). We first give examples of them, compare them with…

In this article we have derived the minimum order of an odd regular graph such that the graph has no matching. We have observed that how it is different from the case of even regular graphs. We have checked the consistency of the derived…

Stimulated by odd-bipartite and even-bipartite hypergraphs, we define odd-bipartite (weakly odd-bipartie) and even-bipartite (weakly even-bipartite) tensors. It is verified that all even order odd-bipartite tensors are irreducible tensors,…

We bound the number of minimal hypergraph transversals that arise in tri-partite 3-uniform hypergraphs, a class commonly found in applications dealing with data. Let H be such a hypergraph on a set of vertices V. We give a lower bound of…

We study minimum degree conditions for which a graph with given odd girth has a simple structure. For example, the classical work of Andr\'asfai, Erd\H os, and S\'os implies that every $n$-vertex graph with odd girth $2k+1$ and minimum…

Consider a graph with $n$ vertices where the shortest odd cycle is of length $>2k+1$. We revisit two known results about such graphs: (I) Such a graph is almost bipartite, in the sense that it can be made bipartite by removing from it…

For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its…

The incidence matrix of a graph is totally unimodular if and only if the graph is bipartite, i.e., it contains no odd cycles. We extend the characterization of total unimodularity to hypergraphs whose hyperedges of size at least four are…

An odd (resp. even) subgraph in a multigraph is its subgraph in which every vertex has odd (resp. even) degree. We say that a multigraph can be decomposed into two odd subgraphs if its edge set can be partitioned into two sets so that both…

In this paper, we study oriented bipartite graphs. In particular, we introduce "bitransitive" graphs. Several characterizations of bitransitive bitournaments are obtained. We show that bitransitive bitounaments are equivalent to acyclic…