Related papers: Eigenvalues and factors: a survey

An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $H$ such that for each vertex $v \in V(G)$, $d_H(v)$ is odd and $1\le d_H(v) \le b$. Let $\lambda_3(G)$ be the third largest eigenvalue of the adjacency matrix of $G$. For positive…

Unigraphs are graphs uniquely determined by their own degree sequence up to isomorphism. There are many subclasses of unigraphs such as threshold graphs, split matrogenic graphs, matroidal graphs, and matrogenic graphs. Unigraphs and these…

In 1977, Hoffman gave a characterization of graphs with smallest eigenvalue at least $-2$. In this paper we generalize this result to graphs with smaller smallest eigenvalue. For the proof, we use a combinatorial object named Hoffman graph,…

The idea of implicit representation of graphs was introduced in [S. Kannan, M. Naor, S. Rudich, Implicit representation of graphs, SIAM J. Discrete Mathematics, 5 (1992) 596--603] and can be defined as follows. A representation of an…

Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the…

We give a necessary and sufficient condition for a cubic graph to be Hamiltonian by analyzing Eulerian tours in certain spanning subgraphs of the quartic graph associated with the cubic graph by 1-factor contraction. This correspondence is…

For a simple graph $G=(V,E)$ with eigenvalues of the adjacency matrix $\lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}$, the energy of the graph is defined by $E(G)=\sum_{j=1}^{n}|\lambda_{j}|$. Myriads of papers have been published in…

In this paper, we continue the classification work done in the first paper of the same name. With careful modifications of our previous approach, we are able to deduce (with two notable exceptions) which members of the previously introduced…

For a fixed graph H with t vertices, an H-factor of a graph G with n vertices, where t divides n, is a collection of vertex disjoint (not necessarily induced) copies of H in G covering all vertices of G. We prove that for a fixed tree T on…

The optimal cube factor of a graph, a special kind of component factor, is first introduced. Furthermore, the optimal cube factors of Fibonacci and matchable Lucas cubes are studied; and some results on the Padovan sequence and binomial…

A Berge $k$-factor in a hypergraph is a generalization of a $k$-factor in a graph. In this paper, we study the problem of determining the values $k$ such that every $\lambda$-edge-connected $r$-regular hypergraph $\HH$ with $k|V(\HH)|$ even…

The aim of this paper is to present a few versions of the Riemann-Hurwitz formula for a regular branched covering of graphs. By a graph, we mean a finite connected multigraph. The genus of a graph is defined as the rank of the first…

In a signed graph $G$, an induced subgraph is called a negative clique if it is a complete graph and all of its edges are negative. In this paper, we give the characteristic polynomials and the eigenvalues of some signed graphs having…

Let d = (d1, d2, ..., dn) be a vector of non-negative integers with even sum. We prove some basic facts about the structure of a random graph with degree sequence d, including the probability of a given subgraph or induced subgraph.…

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the…

A spanning subgraph $H$ of a graph $G$ is called a $P_{\geq k}$-factor of $G$ if every component of $H$ is isomorphic to a path of order at least $k$, where $k\geq2$ is an integer. A graph $G$ is called a $(P_{\geq k},l)$-factor critical…

Assume that $G$ is a finite group. For every $a, b \in\mathbb N,$ we define a graph $\Gamma_{a,b}(G)$ whose vertices correspond to the elements of $G^a\cup G^b$ and in which two tuples $(x_1,\dots,x_a)$ and $(y_1,\dots,y_b)$ are adjacent if…

A (3,4)-biregular bigraph G is a bipartite graph where all vertices in one part have degree 3 and all vertices in the other part have degree 4. A path factor of G is a spanning subgraph whose components are nontrivial paths. We prove that a…

Factor analysis aims to describe high dimensional random vectors by means of a small number of unknown common factors. In mathematical terms, it is required to decompose the covariance matrix $\Sigma$ of the random vector as the sum of a…

In this paper, we prove that the 2-factor polynomial, an invariant of a planar trivalent graph with a perfect matching, counts the number of 2- factors that contain the the perfect matching as a subgraph. Consequently, we show that the…