Constructing a covering triangulation by means of a nowhere-zero dual flow

Tutte conjectured that every graph with no isthmus can be provided with an integral nowhere-zero flow with no absolute value greater than k=5. As yet the result is established for k=6, and it is used for proving that the existence of a triangular imbedding of a graph G in a surface S implies the existence of a triangular imbedding of G_(m) in a surface $\text{S}\text{?}$ with the same orientability characteristic as S. G_(m) stands for the composition of G by an independent set of m vertices.     

Graphs determined by their generalized characteristic polynomials

For a given graph G with (0, 1)-adjacency matrix A_G, the generalized characteristic polynomial of G is defined to be ?_G=?_G(λ,t)=det(λI-(A_G-tD_G)), where I is the identity matrix and D_G is the diagonal degree matrix of G. In this paper, we are mainly concerned with the problem of characterizing a given graph G by its generalized characteristic polynomial ?_G. We show that graphs with the same generalized characteristic polynomials have the same degree sequence, based on which, a unified approach is proposed to show that some families of graphs are characterized by ?_G. We also provide a method for constructing graphs with the same generalized characteristic polynomial, by using GM-switching.     

Triangular line graphs and word sense disambiguation

Linguists often represent the relationships between words in a collection of text as an undirected graph G=(V,E), where V is the vocabulary and vertices are adjacent in G if and only if the words that they represent co-occur in a relevant pattern in the text. Ideally, the words with similar meanings give rise to the vertices of a component of the graph. However, many words have several distinct meanings, preventing components from characterizing distinct semantic fields. This paper examines how the structural properties of triangular line graphs motivate the use of a clustering coefficient on the triangular line graph, thereby helping to identify polysemous words. The triangular line graph of G, denoted by T(G), is the subgraph of the line graph of G where two vertices are adjacent if the corresponding edges in G belong to a K₃.     

On Integral Sum Graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v) = f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K₃ has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous relative results are also presented.     

The genus of the Cartesian product of two graphs

Upper and lower bounds are given for the genus, γ(G₁ × G₂), of the Cartesian product of arbitrary graphs G₁ and G₂, in terms of the genera γ(G₁) and γ(G₂). These bounds are then used to obtain asymptotic results for the cases in which G₁ and G₂ are both regular complete k-partite graphs.     

Graphs without K4 and well-quasi-ordering

It is proved that given an infinite sequence G₁, G₂, G₃,…, of series-parallel graphs there are indices i < j such that G_j contains an induced subgraph contractable onto G_i. An example is given showing that for planar graphs the preceding theorem fails.     

On integral sum graphs

A graph G is said to be an integral sum graph if its nodes can be given a labeling f with distinct integers, so that for any two distinct nodes u and v of G, uv is an edge of G if and only if f(u)+f(v)=f(w) for some node w in G. A node of G is called a saturated node if it is adjacent to every other node of G. We show that any integral sum graph which is not K₃ has at most two saturated nodes. We determine the structure for all integral sum graphs with exactly two saturated nodes, and give an upper bound for the number of edges of a connected integral sum graph with no saturated nodes. We introduce a method of identification on constructing new connected integral sum graphs from given integral sum graphs with a saturated node. Moreover, we show that every graph is an induced subgraph of a connected integral sum graph. Miscellaneous related results are also presented.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

Neighborhood perfect graphs

The notion of neighborhood perfect graphs is introduced here as follows. Let G be a graph, α_N(G) denote the maximum number of edges such that no two of them belong to the same subgraph of G induced by the (closed) neighborhood of some vertex; let ϱ_N(G) be the minimum number of vertices whose neighborhood subgraph cover the edge set of G. Then G is called neighborhood perfect if ϱ_N(G′) = α_N(G′) holds for every induced subgraph G′ of G. It is expected that neighborhood perfect graphs are perfect also in the sense of Berge. We characterized here those chordal graphs which are neighborhood perfect. In addition, an algorithm to computer ϱ_N(G) = α_N(G) is given for interval graphs.     

Cayley graph expanders and groups of finite width

We present new infinite families of expander graphs of vertex degree 4, which is the minimal possible degree for Cayley graph expanders. Our first family defines a tower of coverings (with covering indices equal to 2) and our second family is given as Cayley graphs of finite groups with very short presentations with only two generators and four relations. Both families are based on particular finite quotients of a group G of infinite upper triangular matrices over the ring .We present explicit vector space bases for the finite abelian quotients of the lower exponent-2 groups of G by upper triangular subgroups and prove a particular 3-periodicity of these quotients. We also conjecture that the group G has finite width 3 and finite average width 8/3.     

Quotients of connected regular graphs of even degree

This paper introduces a method of listing all nonequivalent quotients of any connected regular graph of even degree with a given 2-factorization. The method is based on the characterization of connected 2d-regular graphs as Schreier coset graphs given by Gross (J. Combin. Theory Ser. B22 (1977), 227–232). Various representations of a given graph with a fixed 2-factorization are also investigated. The work is related to graph imbedding theory, particularly to voltage and current graphs.     

On complete intersection toric ideals of graphs

We study the graphs G for which their toric ideals I _G are complete intersections. In particular, we prove that for a connected graph G such that I _G is a complete intersection all of its blocks are bipartite except for at most two. We prove that toric ideals of graphs which are complete intersections are circuit ideals. In this case, the generators of the toric ideal correspond to even cycles of G except of at most one generator, which corresponds to two edge disjoint odd cycles joint at a vertex or with a path. We prove that the blocks of these graphs satisfy the odd cycle condition. Finally, we characterize all complete intersection toric ideals of graphs which are normal.     

A class of h-perfect graphs

A graph is said to be h-perfect if the convex hull of its independent sets is defined by the constraints corresponding to cliques and odd holes, and the nonnegativity constraints. Series-parallel graphs and perfect graphs are h-perfect. The purpose of this paper is to extend the class of graphs known to be h-perfect. Thus, given a graph which is the union of a bipartite graph G₁ and a graph G₂ having exactly two common nodes a and b, and no edge in common, we prove that G is h-perfect if so is the graph obtained from G by replacing G₁ by an a-b chain (the length of which depends on G₁). This result enables us to prove that the graph obtained by substituting bipartite graphs for edges of a series-parallel graph is h-perfect, and also that the identification of two nodes of a bipartite graph yields an h-perfect graph (modulo a reduction which preserves h-perfection).     

On the edge-connectivity and restricted edge-connectivity of a product of graphs

The product graph G_m*G_p of two given graphs G_m and G_p was defined by Bermond et al. [Large graphs with given degree and diameter II, J. Combin. Theory Ser. B 36 (1984) 32-48]. For this kind of graphs we provide bounds for two connectivity parameters (λ and λ^′, edge-connectivity and restricted edge-connectivity, respectively), and state sufficient conditions to guarantee optimal values of these parameters. Moreover, we compare our results with other previous related ones for permutation graphs and cartesian product graphs, obtaining several extensions and improvements. In this regard, for any two connected graphs G_m, G_p of minimum degrees δ(G_m), δ(G_p), respectively, we show that λ(G_m*G_p) is lower bounded by both δ(G_m)+λ(G_p) and δ(G_p)+λ(G_m), an improvement of what is known for the edge-connectivity of G_m×G_p.     

Orientable imbedding of line graphs

In this paper we compute the orientable genus of the line graph of a graph G, when G is a tree and a 2-edge connected graph, all the vertices of which have their degrees equal to 2, 3, 6, or 11 modulo 12, and either G can be imbedded with triangular faces only or G is a bipartite graph which can be imbedded with squares only as faces. In the other cases, we give an upper bound of the genus of line graphs. In this way, we solve the question of the Hamiltonian genus of the complete graph K_n, for every n ≥ 3.     

On cubical graphs

It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of n-dimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of those graphs G, called cubical graphs (not to be confused with cubic graphs, those graphs for which all vertices have degree 3), which can be embedded into an n-dimensional cube. A basic technique used is the investigation of graphs which are critically nonembeddable, i.e., which can not be embedded but all of whose subgraphs can be embedded.     

Kirchhoff index of composite graphs

Let G₁+G₂, G₁°G₂ and G₁{G₂} be the join, corona and cluster of graphs G₁ and G₂, respectively. In this paper, Kirchhoff index formulae of these composite graphs are given.     

Tian’s invariant of the Grassmann manifold

We prove that Tian’s invariant on the complex Grassmann manifold G _p,q(?)is equal to 1/(p+ q).The method introduced here uses a Lie group of holomorphic isometries which operates transitively on the considered manifolds and a natural imbedding of (?¹ (?))^p in G _p,q (?).     

On the 2-rainbow domination in graphs

The concept of 2-rainbow domination of a graph G coincides with the ordinary domination of the prism G□K₂. In this paper, we show that the problem of deciding if a graph has a 2-rainbow dominating function of a given weight is NP-complete even when restricted to bipartite graphs or chordal graphs. Exact values of 2-rainbow domination numbers of several classes of graphs are found, and it is shown that for the generalized Petersen graphs GP(n,k) this number is between ⌈4n/5⌉ and n with both bounds being sharp.     

Laplacian integral graphs in S(a, b)

Let G_n,m be the family of graphs with n vertices and m edges, when n and m are previously given. It is well-known that there is a subset of G_n,m constituted by graphs G such that the vertex connectivity, the edge connectivity, and the minimum degree are all equal. In this paper, S(a, b)-classes of connected (a, b)-linear graphs with n vertices and m edges are described, where m is given as a function of a,b∈N/2. Some of them have extremal graphs for which the equalities above are extended to algebraic connectivity. These graphs are Laplacian integral although they are not threshold graphs. However, we do build threshold graphs in S(a, b).