Character degree graphs with no complete vertices

Let Γ be a graph in which each vertex is non-adjacent to another different one. We show that, if G is a finite solvable group with abelian Fitting subgroup and with character degree graph $\Gamma (G)=\Gamma$, then G is a direct product of subgroups having a disconnected character degree graph. In particular, Γ is a join of disconnected graphs. We deduce also that solvable groups with abelian Fitting subgroup have a character degree graph with diameter at most 2.     

Character degree graphs and normal subgroups

We consider the degrees of those irreducible characters of a group whose kernels do not contain a given normal subgroup . We show that if and is not perfect, then the common-divisor graph on this set of integers has at most two connected components. Also, if is solvable, we obtain bounds on the diameters of the components of this graph and, in the disconnected case, we study the structure of and of .

     

Delaunay graphs are almost as good as complete graphs

LetS be any set ofN points in the plane and let DT(S) be the graph of the Delaunay triangulation ofS. For all pointsa andb ofS, letd(a, b) be the Euclidean distance froma tob and let DT(a, b) be the length of the shortest path in DT(S) froma tob. We show that there is a constantc (≤((1+√5)/2) π≈5.08) independent ofS andN such that $$\frac{{DT(a,b)}}{{d(a,b)}}< c.$$      

Trees that are sphere-of-influence graphs

Given a collection of points in the plane, an open (closed) disk is drawn about each point with radius equal to the smallest distance from that point to any other in the collection. The sphere-of-influence graph (closed-sphere-of-influence graph) is the intersection graph of these disks. Any graph isomorphic to such a graph is a SIG (CSIG). We characterize trees that are SIGs and CSIGs, and obtain a bound on the number of edges in triangle-free SIGs and CSIGs.     

Unoriented Laplacian maximizing graphs are degree maximal

A connected graph is said to be unoriented Laplacian maximizing if the spectral radius of its unoriented Laplacian matrix attains the maximum among all connected graphs with the same number of vertices and the same number of edges. A graph is said to be threshold (maximal) if its degree sequence is not majorized by the degree sequence of any other graph (and, in addition, the graph is connected). It is proved that an unoriented Laplacian maximizing graph is maximal and also that there are precisely two unoriented Laplacian maximizing graphs of a given order and with nullity 3. Our treatment depends on the following known characterization: a graph G is threshold (maximal) if and only if for every pair of vertices u,v of G, the sets N(u)?{v},N(v)?{u}, where N(u) denotes the neighbor set of u in G, are comparable with respect to the inclusion relation (and, in addition, the graph is connected). A conjecture about graphs that maximize the unoriented Laplacian matrix among all graphs with the same number of vertices and the same number of edges is also posed.     

Vertex-transitive graphs that are not Cayley graphs. II

The Petersen graph on 10 vertices is the smallest example of a vertex-transitive graph that is not a Cayley graph. In 1983, D. Marus˘ic˘ asked, “For what values of n does there exist such a graph on n vertices?” We give several new constructions of families of vertex-transitive graphs that are not Cayley graphs and complete the proof that, if n is divisible by p² for some prime p, then there is a vertex-transitive graph on n vertices that is not a Cayley graph unless n is p², p³, or 12. © 1996 John Wiley & Sons, Inc.     

A class of finite groups with complete character degree graphs

In this paper, we construct a new class of finite groups whose common divisor graphs are complete graphs, while there is no prime dividing all the nontrivial degrees.     

The relative degree and large complete minors in infinite graphs

Finite graphs of large minimum degree have large complete (topological) minors. We propose a new and very natural notion, the relative degree of an end, which makes it possible to extend this fact to locally finite graphs and to graphs with countably many ends. We conjecture the extension to be true for all infinite graphs.     

Graphs that are not complete pluripolar

Let be domains in . Under very mild conditions on we show that there exist holomorphic functions , defined on with the property that is nowhere extendible across , while the graph of over is not complete pluripolar in . This refutes a conjecture of Levenberg, Martin and Poletsky (1992).

     

拟正则完全二部图的局部最可靠性

用P(G，ρ)表示顶点完全可靠，而边则以概率ρ∈(0，1)相互独立地出现故障的图G的全终端不可靠度，即G因边故障而变得不连通的概率．本文证明了边故障率ρ充分小时，拟正则完全二部图在具有相同点数和相同边数的图类中是惟一具有最小全终端不可靠度的图．     

Distance-regular graphs of large diameter that are completely regular clique graphs

A connected graph is said to be a completely regular clique graph with parameters (s, c), \(s, c \in {\mathbb {N}}\), if there is a collection \(\mathcal {C}\) of completely regular cliques of size \(s+1\) such that every edge is contained in exactly c members of \(\mathcal {C}\). It is known that many families of distance-regular graphs are completely regular clique graphs. In this paper, we determine completely regular clique graph structures, i.e., the choices of \(\mathcal {C}\), of all known families of distance-regular graphs with unbounded diameter. In particular, we show that all distance-regular graphs in this category are completely regular clique graphs except the Doob graphs, the twisted Grassmann graphs and the Hermitean forms graphs. We also determine parameters (s, c); however, in a few cases we determine only s and give a bound on the value c. Our result is a generalization of a series of works by J. Hemmeter and others who determined distance-regular graphs in this category that are bipartite halves of bipartite distance-regular graphs.     

An improved upper bound on the maximum degree of terminal-pairable complete graphs

A graph $G$ is terminal-pairable with respect to a demand (loopless) multigraph $D$ on the same vertex set as $G$, if there exist edge-disjoint paths joining the end vertices of every demand edge of $D$. In this short note, we improve the upper bound on the largest $\Delta \left(n\right)$ with the property that the complete graph on $n$ vertices is terminal-pairable with respect to any demand multigraph of maximum degree at most $\Delta \left(n\right)$. This disproves a conjecture originally stated by Csaba, Faudree, Gyárfás, Lehel and Schelp.     

Affine semigroup rings that are complete intersections

This paper presents a result concerning the structure of affine semigroup rings that are complete intersections. It generalizes to arbitrary dimensions earlier results for semigroups of dimension less than four. The proof depends on a decomposition theorem for mixed dominating matrices.

     

Planar graphs with maximum degree 4 are strongly 19-edge-colorable

A strong edge-coloring of a graph is a proper edge-coloring such that edges at distance at most 2 receive different colors. It is known that every planar graph has a strong edge-coloring by using at most $4\Delta +4$ colors, where $\Delta$ denotes the maximum degree of the graph. In this paper, we will show that 19 colors are enough to color a planar graph with maximum degree 4.     

Sets that are connected in two random graphs

–We consider two random graphs G₁, G₂, both on the same vertex set. We ask whether there is a non‐trivial set of vertices S, so that S induces a connected subgraph both in G₁ and in G₂. We determine the threshold for the appearance of such a subset, as well as the size of the largest such subset. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 498–512, 2014     

Decompositions of complete graphs and complete bipartite graphs into isomorphic supersubdivision graphs

A graph H is called a supersubdivison of a graph G if H is obtained from G by replacing every edge uv of G by a complete bipartite graph K_2,m (m may vary for each edge) by identifying u and v with the two vertices in K_2,m that form one of the two partite sets. We denote the set of all such supersubdivision graphs by SS(G). Then, we prove the following results.

1. Each non-trivial connected graph G and each supersubdivision graph HSS(G) admits an α-valuation. Consequently, due to the results of Rosa (in: Theory of Graphs, International Symposium, Rome, July 1966, Gordon and Breach, New York, Dunod, Paris, 1967, p. 349) and El-Zanati and Vanden Eynden (J. Combin. Designs 4 (1996) 51), it follows that complete graphs K_2cq+1 and complete bipartite graphs K_mq,nq can be decomposed into edge disjoined copies of HSS(G), for all positive integers m,n and c, where q=|E(H)|.

2. Each connected graph G and each supersubdivision graph in SS(G) is strongly n-elegant, where n=|V(G)| and felicitous.

3. Each supersubdivision graph in EASS(G), the set of all even arbitrary supersubdivision graphs of any graph G, is cordial.

Further, we discuss a related open problem.     

Degree complete graphs

Ryser [Combinatorial Mathematics, Carus Mathematical Monograph, vol. 14, Wiley, New York, 1963] introduced a partially ordered relation ‘?’ on the nonnegative integral vectors. It is clear that if S=(s₁,s₂,…,s_n) is an out-degree vector of an orientation of a graph G with vertices 1,2,…,n, then

(Π)     

Lattice complete graphs

We study the properties of graphs that can be placed in a rectangular lattice so that all vertices located in the same (horizontal or vertical) row be adjacent. Some criterion is formulated for an arbitrary graph to be in the specified class.