Solution to a problem of C. D. Godsil regarding bipartite graphs with unique perfect matching

We give the solution to the following question of C. D. Godsil[2]: Among the bipartite graphsG with a unique perfect matching and such that a bipartite graph obtains when the edges of the matching are contracted, characterize those having the property thatG ⁺G, whereG ⁺ is the bipartite multigraph whose adjacency matrix,B ⁺, is diagonally similar to the inverse of the adjacency matrix ofG put in lower-triangular form. The characterization is thatG must be obtainable from a bipartite graph by adding, to each vertex, a neighbor of degree one. Our approach relies on the association of a directed graph to each pair (G, M) of a bipartite graphG and a perfect matchingM ofG.     

Interval Non‐edge‐Colorable Bipartite Graphs and Multigraphs

An edge‐coloring of a graph G with colors is called an interval t‐coloring if all colors are used, and the colors of edges incident to any vertex of G are distinct and form an interval of integers. In 1991, Erd?s constructed a bipartite graph with 27 vertices and maximum degree 13 that has no interval coloring. Erd?s's counterexample is the smallest (in a sense of maximum degree) known bipartite graph that is not interval colorable. On the other hand, in 1992, Hansen showed that all bipartite graphs with maximum degree at most 3 have an interval coloring. In this article, we give some methods for constructing of interval non‐edge‐colorable bipartite graphs. In particular, by these methods, we construct three bipartite graphs that have no interval coloring, contain 20, 19, 21 vertices and have maximum degree 11, 12, 13, respectively. This partially answers a question that arose in [T.R. Jensen, B. Toft, Graph coloring problems, Wiley Interscience Series in Discrete Mathematics and Optimization, 1995, p. 204]. We also consider similar problems for bipartite multigraphs.     

Eulerian subgraphs in 3‐edge‐connected graphs and Hamiltonian line graphs

In this paper, we show that if G is a 3‐edge‐connected graph with and , then either G has an Eulerian subgraph H such that , or G can be contracted to the Petersen graph in such a way that the preimage of each vertex of the Petersen graph contains at least one vertex in S. If G is a 3‐edge‐connected planar graph, then for any , G has an Eulerian subgraph H such that . As an application, we obtain a new result on Hamiltonian line graphs. © 2003 Wiley Periodicals, Inc. J Graph Theory 42: 308–319, 2003     

Constructions of covering triangulations with folds

Let G be a graph with a known triangular embedding in a surface S, and consider G_(m), the composition of G with an independant set of order m. The purpose of this paper is to construct a triangular embedding of G_(m) into a surface by using a covering triangulation with folds. We make the construction for three cases. One of them is used for proving that G_(m) can be triangularly embedded into a surface if G is an Eulerian graph which can be triangularly embedded into a surface S with the same orientability characteristic as .     

A Fiedler-like theory for the perturbed Laplacian

The perturbed Laplacian matrix of a graph G is defined as DL = D?A, where D is any diagonal matrix and A is a weighted adjacency matrix of G. We develop a Fiedler-like theory for this matrix, leading to results that are of the same type as those obtained with the algebraic connectivity of a graph. We show a monotonicity theorem for the harmonic eigenfunction corresponding to the second smallest eigenvalue of the perturbed Laplacian matrix over the points of articulation of a graph. Furthermore, we use the notion of Perron component for the perturbed Laplacian matrix of a graph and show how its second smallest eigenvalue can be characterized using this definition.     

Trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues

The signless Laplacian matrix of a graph G is defined to be the sum of its adjacency matrix and degree diagonal matrix, and its eigenvalues are called Q-eigenvalues of G. A Q-eigenvalue of a graph G is called a Q-main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. In this work, all trees, unicyclic graphs and bicyclic graphs with exactly two Q-main eigenvalues are determined.     

Proof of conjecture involving the second largest signless Laplacian eigenvalue and the index of graphs

Let G = (V, E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the signless Laplacian matrix of G is Q(G) = D(G) + A(G). In [5], Cvetkovi? et al. have given the following conjecture involving the second largest signless Laplacian eigenvalue (q₂) and the index (λ₁) of graph G (see also Aouchiche and Hansen [1]):

     

On the nullity of bipartite graphs

The nullity of a graph is defined to be the multiplicity of the eigenvalue zero in the spectrum of the adjacency matrix of the graph. In this paper, we obtain the nullity set of bipartite graphs of order n, and characterize the bipartite graphs with nullity n-4 and the regular bipartite graphs with nullity n-6.     

Some results on cyclic interval edge colorings of graphs

A proper edge coloring of a graph G with colors is called a cyclic interval t‐coloring if for each vertex v of G the edges incident to v are colored by consecutive colors, under the condition that color 1 is considered as consecutive to color t. We prove that a bipartite graph G of even maximum degree admits a cyclic interval ‐coloring if for every vertex v the degree satisfies either or . We also prove that every Eulerian bipartite graph G with maximum degree at most eight has a cyclic interval coloring. Some results are obtained for ‐biregular graphs, that is, bipartite graphs with the vertices in one part all having degree a and the vertices in the other part all having degree b; it has been conjectured that all these have cyclic interval colorings. We show that all (4, 7)‐biregular graphs as well as all ‐biregular () graphs have cyclic interval colorings. Finally, we prove that all complete multipartite graphs admit cyclic interval colorings; this proves a conjecture of Petrosyan and Mkhitaryan.     

Cutting a graph into two dissimilar halves

Given a graph G and a subset S of the vertex set of G, the discrepancy of S is defined as the difference between the actual and expected numbers of the edges in the subgraph induced on S. We show that for every graph with n vertices and e edges, n < e < n(n ? 1)/4, there is an n/2-element subset with the discrepancy of the order of magnitude of \documentclass{article}\pagestyle{empty}\begin{document}$\sqrt {ne}$\end{document} For graphs with fewer than n edges, we calculate the asymptotics for the maximum guaranteed discrepancy of an n/2-element subset. We also introduce a new notion called “bipartite discrepancy” and discuss related results and open problems.     

The preconditioned inverse iteration for hierarchical matrices

The preconditioned inverse iteration is an efficient method to compute the smallest eigenpair of a symmetric positive definite matrix M. Here we use this method to find the smallest eigenvalues of a hierarchical matrix. The storage complexity of the data‐sparse ‐matrices is almost linear. We use ‐arithmetic to precondition with an approximate inverse of M or an approximate Cholesky decomposition of M. In general, ‐arithmetic is of linear‐polylogarithmic complexity, so the computation of one eigenvalue is cheap. We extend the ideas to the computation of inner eigenvalues by computing an invariant subspace S of (M ? μI)² by subspace preconditioned inverse iteration. The eigenvalues of the generalized matrix Rayleigh quotient μ_M(S) are the desired inner eigenvalues of M. The idea of using (M ? μI)² instead of M is known as the folded spectrum method. As we rely on the positive definiteness of the shifted matrix, we cannot simply apply shifted inverse iteration therefor. Numerical results substantiate the convergence properties and show that the computation of the eigenvalues is superior to existing algorithms for non‐sparse matrices.Copyright © 2012 John Wiley & Sons, Ltd.     

Sparse pseudo‐random graphs are Hamiltonian

In this article we study Hamilton cycles in sparse pseudo‐random graphs. We prove that if the second largest absolute value λ of an eigenvalue of a d‐regular graph G on n vertices satisfies and n is large enough, then G is Hamiltonian. We also show how our main result can be used to prove that for every c >0 and large enough n a Cayley graph X (G,S), formed by choosing a set S of c log⁵ n random generators in a group G of order n, is almost surely Hamiltonian. © 2002 Wiley Periodicals, Inc. J Graph Theory 42: 17–33, 2003     

On the residue of a graph

The residue R of a simple graph G of degree sequence S: d₁ ? d₂ ? …? ? d_n is the number of zeros obtained by the iterative process consisting of deleting the first term d₁ of S, subtracting 1 from the d₁ following ones, and sorting down the new sequence. The depth is the number n - R of steps in this algorithm. We prove here some conjectures given by the computer program GRAFFITI, in particular, .     

The decycling number and maximum genus of cubic graphs

Let and denote the minimum size of a decycling set and maximum genus of a graph G, respectively. For a connected cubic graph G of order n, it is shown that . Applying the formula, we obtain some new results on the decycling number and maximum genus of cubic graphs. Furthermore, it is shown that the number of vertices of a decycling set S in a k‐regular graph G is , where c and are the number of components of and the number of edges in , respectively. Therefore, S is minimum if and only if is minimum. As an application, this leads to a lower bound for of a k‐regular graph G. In many cases this bound may be sharp.     

On conjectures involving second largest signless Laplacian eigenvalue of graphs

Let G=(V,E) be a simple graph. Denote by D(G) the diagonal matrix of its vertex degrees and by A(G) its adjacency matrix. Then the Laplacian matrix of G is L(G)=D(G)-A(G) and the signless Laplacian matrix of G is Q(G)=D(G)+A(G). In this paper we obtain a lower bound on the second largest signless Laplacian eigenvalue and an upper bound on the smallest signless Laplacian eigenvalue of G. In [5], Cvetkovi? et al. have given a series of 30 conjectures on Laplacian eigenvalues and signless Laplacian eigenvalues of G (see also [1]). Here we prove five conjectures.     

Almost all trees share a complete set of immanantal polynomials

Let χ be an irreducible character of the symmetric group S_n. For an n-by-n matrix A = (a_ij), define If G is a graph, let D(G) be the diagonal matrix of its vertex degrees and A(G) its adjacency matrix. Let y and z be independent indeterminates, and define L(G) = yD(G) + zA(G). Suppose t_n is the number of trees on n vertices and s_n is the number of such trees T for which there exists a nonisomorphic tree T? such that d_χ(xl - L(T)) = d_x(xl - L(T?)) for every irreducible character χ of S_n. Then lim_n→∞ S_n/t_n = 1. © 1993 John Wiley & Sons, Inc.     

Eigenvalues and forbidden subgraphs I

Suppose a graph G have n vertices, m edges, and t triangles. Letting λ_n(G) be the largest eigenvalue of the Laplacian of G and μ_n(G) be the smallest eigenvalue of its adjacency matrix, we prove that

     

On an anti‐Ramsey problem of Burr,Erdős,Graham, and T. Sós

Given a graph L, in this article we investigate the anti‐Ramsey number χ_S(n,e,L), defined to be the minimum number of colors needed to edge‐color some graph G(n,e) with n vertices and e edges so that in every copy of L in G all edges have different colors. We call such a copy of L totally multicolored (TMC). In 7 among many other interesting results and problems, Burr, Erd?s, Graham, and T. Sós asked the following question: Let L be a connected bipartite graph which is not a star. Is it true then that In this article, we prove a slightly weaker statement, namely we show that the statement is true if L is a connected bipartite graph, which is not a complete bipartite graph. © 2006 Wiley Periodicals, Inc. J Graph Theory 52: 147–156, 2006     

On 3-colorings of bipartite p-threshold graphs

In this paper some extremal properties of 3-colorings of bipartite complete graphs in the class of all bipartite p-threshold graphs that are uniquely 2-colorable are proved. As a consequence it is shown that the complete bipartite graphs K_{p, p + r} where p ? 2 and 0 ? r < are chromatically unique. A useful result concerning the maximization of a sum of powers of two under certain restrictions, which has an arithmetical interest, is also presented.