Constructing cospectral graphs via a new form of graph product

We present a new method to construct a family of co-spectral graphs. Our method is based on a new type of graph product that we define, the bipartite graph product, which may be of self-interest. Our method is different from existing techniques in the sense that it is not based on a sequence of local graph operations (e.g. Godsil–McKay switching). The explicit nature of our construction allows us, for example, to construct an infinite family of cospectral graphs and provide an easy proof of non-isomorphism. We are also able to characterize fully the spectrum of the cospectral graphs.     

Deza graphs: A generalization of strongly regular graph

We consider the following generalization of strongly regular graphs. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). We introduce several ways to construct Deza graphs, and develop some basic theory. We also list all diameter two Deza graphs which are not strongly regular and have at most 13 vertices.     

The edge-subgraph poset of a graph and the edge reconstruction conjecture

We consider only finite simple graphs in this paper. Earlier we showed that many invariants of a graph can be computed from the isomorphism class of its partially ordered set of distinct unlabeled non-empty induced subgraphs, that is, the subgraphs themselves are not required. In this paper, we consider an analogous problem of reconstructing an arbitrary graph up to isomorphism from its abstract edge-subgraph poset , which we call the -reconstruction problem. We present an infinite family of graphs that are not -reconstructible and show that the edge reconstruction conjecture is true if and only if the graphs in the family are the only graphs that are not -reconstructible. Let be the set of all unlabeled graphs. Let denote the number of homomorphisms from to . Let be a bijection such that for all , we have . We conjecture that is the identity map. Our conjecture is motivated by the homomorphism cancellation results of Lovász. We prove that the conjecture stated above is weaker than the edge reconstruction conjecture.     

The number of edges of radius-invariant graphs

The eccentricity e(υ) of vertex υ is defined as a distance to a farthest vertex from υ. The radius of a graph G is defined as r(G) = {e(u)}. We consider properties of unchanging the radius of a graph under two different situations: deleting an arbitrary edge and deleting an arbitrary vertex. This paper gives the upper bounds for the number of edges in such graphs. Supported by VEGA grant No. 1/0084/08.     

The order of the largest complete minor in a random graph

Let ccl(G) denote the order of the largest complete minor in a graph G (also called the contraction clique number) and let G_n,p denote a random graph on n vertices with edge probability p. Bollobás, Catlin, and Erd?s (Eur J Combin 1 (1980), 195–199) asymptotically determined ccl(G_n,p) when p is a constant. ?uczak, Pittel and Wierman (Trans Am Math Soc 341 (1994) 721–748) gave bounds on ccl(G_n,p) when p is very close to 1/n, i.e. inside the phase transition. We show that for every ε > 0 there exists a constant C such that whenever C/n < p < 1 ‐ ε then asymptotically almost surely ccl(G_n,p) = (1 ± ε)n/ , where b := 1/(1 ‐ p). If p = C/n for a constant C > 1, then ccl(G_n,p) = Θ( ). This extends the results in (Bollobás, Catlin, and P. Erd?s, Eur J Combin 1 (1980), 195–199) and answers a question of Krivelevich and Sudakov (preprint, 2006). © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Symmetrical path-cycle covers of a graph and polygonal graphs

A near-polygonal graph is a graph Γ which has a set C of m-cycles for some positive integer m such that each 2-path of Γ is contained in exactly one cycle in C. If m is the girth of Γ then the graph is called polygonal. We introduce a method for constructing near-polygonal graphs with 2-arc transitive automorphism groups. As special cases, we obtain several new infinite families of polygonal graphs.     

完全五部图的S-整图性研究

应用矩阵的初等变换得到了完全五部图的 Seidel 多项式,并给出了完全五部图是S-整图的一个充分必要条件。进一步刻画了完全正则五部图和两类特殊完全五部图的Seidel谱。     

On the Laplacian energy of a graph

In this paper we consider the energy of a simple graph with respect to its Laplacian eigenvalues, and prove some basic properties of this energy. In particular, we find the minimal value of this energy in the class of all connected graphs on n vertices (n = 1, 2, ...). Besides, we consider the class of all connected graphs whose Laplacian energy is uniformly bounded by a constant α ⩾ 4, and completely describe this class in the case α = 40.     

On the stable set polytope of a series-parallel graph

We give a short proof of Chvátal's conjecture that the nontrivial facets of the stable set polytope of a series-parallel graph all come from edges and odd holes.Research supported in part by the Natural Sciences and Engineering Research Council of Canada and by CP Rail.     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs     

The join of split graphs whose half-strong endomorphisms form a monoid

In this paper, the half-strong endomorphisms of the join of split graphs are investigated. We give the conditions under which the half-strong endomorphisms of the join of split graphs form a monoid.     

The fine structure of octahedron-free graphs

This paper is one of a series of papers in which, for a family L of graphs, we describe the typical structure of graphs not containing any L∈L. In this paper, we prove sharp results about the case L={O₆}, where O₆ is the graph with 6 vertices and 12 edges, given by the edges of an octahedron. Among others, we prove the following results.(a) The vertex set of almost every O₆-free graph can be partitioned into two classes of almost equal sizes, U₁ and U₂, where the graph spanned by U₁ is a C₄-free and that by U₂ is P₃-free.(b) Similar assertions hold when L is the family of all graphs with 6 vertices and 12 edges.(c) If H is a graph with a color-critical edge and χ(H)=p+1, then almost every sH-free graph becomes p-chromatic after the deletion of some s−1 vertices, where sH is the graph formed by s vertex disjoint copies of H.These results are natural extensions of theorems of classical extremal graph theory. To show that results like those above do not hold in great generality, we provide examples for which the analogs of our results do not hold.     

The number of graphs and a random graph with a given degree sequence

We consider the set of all graphs on n labeled vertices with prescribed degrees D = (d₁,…,d_n). For a wide class of tame degree sequences D we obtain a computationally efficient asymptotic formula approximating the number of graphs within a relative error which approaches 0 as n grows. As a corollary, we prove that the structure of a random graph with a given tame degree sequence D is well described by a certain maximum entropy matrix computed from D. We also establish an asymptotic formula for the number of bipartite graphs with prescribed degrees of vertices, or, equivalently, for the number of 0‐1 matrices with prescribed row and column sums. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

The number of defective colorings of graphs on surfaces

A (k, 1)‐coloring of a graph is a vertex‐coloring with k colors such that each vertex is permitted at most 1 neighbor of the same color. We show that every planar graph has at least cρⁿ distinct (4, 1)‐colorings, where c is constant and ρ≈1.466 satisfies ρ³ = ρ² + 1. On the other hand for any ε>0, we give examples of planar graphs with fewer than c(? + ε)ⁿ distinct (4, 1)‐colorings, where c is constant and . Let γ(S) denote the chromatic number of a surface S. For every surface S except the sphere, we show that there exists a constant c′ = c′(S)>0 such that every graph embeddable in S has at least c′2ⁿ distinct (γ(S), 1)‐colorings. © 2010 Wiley Periodicals, Inc. J Graph Theory 28:129‐136, 2011