Small regular graphs having the same path layer matrix*

The path layer matrix of graph G contains quantitative information about all paths in G. The entry (i,j) in this matrix is the number of simple paths in G having initial vertex i and length j. Some new upper bounds for r‐regular graphs with the same path layer matrix are presented for r=4, 5, 6. © 2002 Wiley Periodicals, Inc. J Graph Theory 39: 219–221, 2002; DOI 10.1002/jgt.10005     

Cubic graphs with 62 vertices having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all paths in G. The entry (i,j) in this matrix is the number of simple paths in G having initial vertex v and length j. It was known that there are cubic graphs on 90 vertices having the same path layer matrix (Dobrynin, 1990). A new upper bound for the least order of cubic graphs with the same path layer matrix is presented. © 1993 John Wiley & Sons, Inc.     

Regular graphs having the same path layer matrix

The path layer matrix (or path degree sequence) of a graph G contains quantitative information about all paths in G. Elements (i,j) in this matrix is the number of simple paths in G having initial vertex v, and length j. For every r ≥ 3, pairs of nonisomorphic r-regular graphs having the same path layer matrix are presented.     

Long cycles passing through vertices in k‐connected graphs

In this article, we consider the following problem proposed by Locke and Zhang in 1991: Let G be a k‐connected graph with minimum degree d and X a set of m vertices on a cycle of G. For which values of m and k, with , must G have a cycle of length at least passing through X? Fujisawa and Yamashita solved this problem for the case and in 2008. We provide an affirmative answer to this problem for the case of and .     

Quasi‐random graphs with given degree sequences

It is now known that many properties of the objects in certain combinatorial structures are equivalent, in the sense that any object possessing any of the properties must of necessity possess them all. These properties, termed quasirandom, have been described for a variety of structures such as graphs, hypergraphs, tournaments, Boolean functions, and subsets of Z _n, and most recently, sparse graphs. In this article, we extend these ideas to the more complex case of graphs which have a given degree sequence. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Scale‐free graphs of increasing degree

We study a scale‐free random graph process in which the number of edges added at each step increases. This differs from the standard model in which a fixed number, m, of edges are added at each step. Let f(t) be the number of edges added at step t. In the standard scale‐free model, f(t) = m constant, whereas in this paper we consider f(t) = [t^c],c > 0. Such a graph process, in which the number of edges grows non‐linearly with the number of vertices is said to have accelerating growth. We analyze both an undirected and a directed process. The power law of the degree sequence of these processes exhibits widely differing behavior. For the undirected process, the terminal vertex of each edge is chosen by preferential attachment based on vertex degree. When f(t) = m constant, this is the standard scale‐free model, and the power law of the degree sequence is 3. When f(t) = [t^c],c < 1, the degree sequence of the process exhibits a power law with parameter x = (3 ? c)/(1 ? c). As c → 0, x → 3, which gives a value of x = 3, as in standard scale‐free model. Thus no more slowly growing monotone function f(t) alters the power law of this model away from x = 3. When c = 1, so that f(t) = t, the expected degree of all vertices is t, the vertex degree is concentrated, and the degree sequence does not have a power law. For the directed process, the terminal vertex is chosen proportional to in‐degree plus an additive constant, to allow the selection of vertices of in‐degree zero. For this process when f(t) = m is constant, the power law of the degree sequence is x = 2 + 1/m. When f(t) = [t^c], c > 0, the power law becomes x = 1 + 1/(1 + c), which naturally extends the power law to [1,2]. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 396–421, 2011     

Tight bounds for the vertices of degree k in minimally k‐connected graphs

For minimally k‐connected graphs on n vertices, Mader proved a tight lower bound for the number of vertices of degree k in dependence on n and k. Oxley observed 1981 that in many cases a considerably better bound can be given if is used as additional parameter, i.e. in dependence on m, n, and k. It was left open to determine whether Oxley's more general bound is best possible. We show that this is not the case, but give a closely related bound that deviates from a variant of Oxley's long‐standing one only for small values of m. We prove that this new bound is best possible. The bound contains Mader's bound as special case.     

Density of the circular chromatic numbers of series‐parallel graphs

In this paper we show that every simple cubic graph on n vertices has a set of at least ? n/4 ? disjoint 2‐edge paths and that this bound is sharp. Our proof provides a polynomial time algorithm for finding such a set in a simple cubic graph. © 2003 Wiley Periodicals, Inc. J Graph Theory 45: 57–79, 2003     

Enumerating the orientable 2‐cell imbeddings of complete bipartite graphs

A formula is developed for the number of congruence classes of 2‐cell imbeddings of complete bipartite graphs in closed orientable surfaces. © 1999 John Wiley & Sons, Inc. J Graph Theory 30: 77–90, 1999     

List‐Coloring the Squares of Planar Graphs without 4‐Cycles and 5‐Cycles

Let G be a planar graph without 4‐cycles and 5‐cycles and with maximum degree . We prove that . For arbitrarily large maximum degree Δ, there exist planar graphs of girth 6 with . Thus, our bound is within 1 of being optimal. Further, our bound comes from coloring greedily in a good order, so the bound immediately extends to online list‐coloring. In addition, we prove bounds for ‐labeling. Specifically, and, more generally, , for positive integers p and q with . Again, these bounds come from a greedy coloring, so they immediately extend to the list‐coloring and online list‐coloring variants of this problem.     

A Characterization of the degree sequences of 2‐trees

A graph G is a 2‐tree if G = K₃, or G has a vertex v of degree 2, whose neighbors are adjacent, and G/ v is a 2‐ tree. A characterization of the degree sequences of 2‐trees is given. This characterization yields a linear‐time algorithm for recognizing and realizing degree sequences of 2‐trees. © 2008 Wiley Periodicals, Inc. J Graph Theory 58:191‐209, 2008     

On approximating the nearest Ω‐stable matrix

In this paper, we consider the problem of approximating a given matrix with a matrix whose eigenvalues lie in some specific region Ω of the complex plane. More precisely, we consider three types of regions and their intersections: conic sectors, vertical strips, and disks. We refer to this problem as the nearest Ω‐stable matrix problem. This includes as special cases the stable matrices for continuous and discrete time linear time‐invariant systems. In order to achieve this goal, we parameterize this problem using dissipative Hamiltonian matrices and linear matrix inequalities. This leads to a reformulation of the problem with a convex feasible set. By applying a block coordinate descent method on this reformulation, we are able to compute solutions to the approximation problem, which is illustrated on some examples.