New bounds on the energy of a graph

Optimality of a linear inequality in finitely many graph invariants is defined through a geometric approach. For a fixed number of graph vertices, consider all the tuples of values taken by the invariants on a selected class of graphs. Then form the polytope which is the convex hull of all these tuples. By definition, the optimal linear inequalities correspond to the facets of this polytope. They are finite in number, are logically independent, and generate precisely all the linear inequalities valid on the class of graphs. The computer system GraPHedron, developed by some of the authors, is able to produce experimental data about such inequalities for a “small” number of vertices. It greatly helps in conjecturing optimal linear inequalities, which are then hopefully proved for any number of vertices. Two examples are investigated here for the class of connected graphs. First, all the optimal linear inequalities for the stability number and the number of edges are obtained. To this aim, a problem of Ore (1962) related to the Turán Theorem (1941) is solved. Second, several optimal inequalities are established for three invariants: the maximum degree, the irregularity, and the diameter. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

It is known that random directed graphs undergo a phase transition around the point . Earlier, Łuczak and Seierstad have established that as when , the asymptotic probability that the strongly connected components of a random directed graph are only cycles and single vertices decreases from 1 to 0 as goes from to . By using techniques from analytic combinatorics, we establish the exact limiting value of this probability as a function of and provide more statistical insights into the structure of a random digraph around, below and above its transition point. We obtain the limiting probability that a random digraph is acyclic and the probability that it has one strongly connected complex component with a given difference between the number of edges and vertices (called excess). Our result can be extended to the case of several complex components with given excesses as well in the whole range of sparse digraphs. Our study is based on a general symbolic method which can deal with a great variety of possible digraph families, and a version of the saddle point method which can be systematically applied to the complex contour integrals appearing from the symbolic method. While the technically easiest model is the model of random multidigraphs, in which multiple edges are allowed, and where edge multiplicities are sampled independently according to a Poisson distribution with a fixed parameter , we also show how to systematically approach the family of simple digraphs, where multiple edges are forbidden, and where 2‐cycles are either allowed or not. Our theoretical predictions are supported by numerical simulations when the number of vertices is finite, and we provide tables of numerical values for the integrals of Airy functions that appear in this study.

CONTENTS Introduction § 1. Basic definitions and auxiliary assertions § 2. Graphs with a given number of vertices § 3. Auxiliary assertions concerning graphs with a given number of vertices and edges § 4. Graphs with a given number of vertices and edges References

The split-and-drift random graph, a null model for speciation

Barnette’s conjecture is an unsolved problem in graph theory. The problem states that every 3-regular (cubic), 3-connected, planar, bipartite (Barnette) graph is Hamiltonian. Partial results have been derived with restrictions on the number of vertices, several properties of face-partitions and dual graphs of Barnette graphs, while some studies focus just on structural characterizations of Barnette graphs. Noting that spider web graphs are a subclass of Annular Decomposable Barnette (ADB graphs) graphs and are Hamiltonian, we study ADB graphs and their annular-connected subclass (ADB-AC graphs). We show that ADB-AC graphs can be generated from the smallest Barnette graph (B 0) using recursive edge operations. We derive several conditions assuring the existence of Hamiltonian cycles in ADB-AC graphs without imposing restrictions on the number of vertices, face size or any other constraints on the face partitions. We show that there can be two types of annuli in ADB-AC graphs, ring annuli and block annuli. Our main result is, ADB-AC graphs having non-singular sequences of ring annuli are Hamiltonian.

Distance-balanced graphs: Symmetry conditions

Efficient dissemination of information remains a central challenge for all types of networks. There are two ways to handle this issue. One way is to compress the amount of data being transferred and the second way is to minimize the delay of information distribution. Well-received approaches used in the second way either design efficient algorithms or implement reliable network architectures with optimal dissemination time. Among the well-known network architectures, the Knödel graph can be considered a suitable candidate for the problem of information dissemination. The Knödel graph Wd,n is a regular graph, of an even order n and degree d, 1 ≤ d ≤ ⌊log2 n⌋. The Knödel graph was introduced by W. Knödel almost four decades ago as network architecture with good properties in terms of broadcasting and gossiping in interconnected networks. Although the Knödel graph has a highly symmetric structure, its diameter is only known for, Wd2d. Recently, the general upper and lower bounds on diameter and broadcast time of the Knödel graph have been presented.

We develop new structure theory for highly regular combinatorial objects, including Steiner designs, strongly regular graphs, and coherent configurations. As applications, we make progress on old problems in algebraic combinatorics and the theory of permutation groups, and break decades-old barriers on the complexity of the algorithmic Graph Isomorphism problem. A central aspect of our structural contributions is the discovery of clique geometries in regular structures. A second aspect is bounds on the rate of expansion of small sets. In the case of Steiner designs, we give a $n^{O(\log n)}$ bound on the number of automorphisms where $n$ is the number of points. This result is nearly optimal in two ways: it essentially matches the number of automorphisms in affine or projective space, and we show that the bound does not extend to the broader class of balanced incomplete block designs. The line-graphs of Steiner designs are strongly regular graphs, and in fact are one of the cases of Neumaier's classification of strongly regular graphs. We bound the number of reconstructions of a Steiner design from its line-graph in order to apply our automorphism bound for Steiner designs to this class of strongly regular graphs, and show that this class of strongly regular graphs has at most $\exp(\tilde{O}(v^{1/14}))$ automorphisms, where $v$ is the number of vertices and the $\tilde{O}$ hides polylogarithmic factors. We give an $\exp(\tilde{O}(1 + \lambda/\mu))$ bound on the number of automorphisms of any nontrivial $\SR(v,\rho,\lambda,\mu)$ strongly regular graph. (Here, $v$ is the number of vertices, $\rho$ is the valency, and $\lambda$ and $\mu$ are the number of common neighbors of a pair of adjacent and nonadjacent vertices, respectively.) As a consequence, we obtain a quasipolynomial bound on the number of automorphisms when $\rho = \Omega(v^{5/6})$. In further study of the structure of the automorphism groups of $\SR(v,\rho,\lambda,\mu)$ graphs, we find a $\Gamma_\mu$ subgroup of index $v^{O(\log v)}$ (i.e., a subgroup of index $v^{O(\log v)}$ for which all composition factors are subgroups of $S_{\mu}$) with known exceptions. In combination with our bound on the number of automorphisms and an earlier bound due to Spielman, we find a $\Gamma_d$ subgroup of the automorphism group of index $v^d$, where $d = \tilde{O}(v^{1/5})$, again with known exceptions. We classify the primitive coherent configurations with not less than $\exp(\tilde{O}(v^{1/3}))$ automorphisms, where $v$ is the number of vertices. As a corollary to our combinatorial classification result, we infer a classification of large primitive permutation groups, previously known only through the Classification of Finite Simple Groups. As a consequence of the combinatorial structure underlying our bounds for the automorphism groups, we give corresponding bounds for the time-complexity of deciding isomorphism. When we bound the order of the automorphism group, our time-complexity bounds are identical to the bounds on the order. From our study of the composition factors of automorphism groups of strongly regular graphs, we obtain a $v^{\mu + O(\log v)}$ and a $\exp(\tilde{O}(v^{1/5}))$ bound on the time-complexity of deciding isomorphism of strongly regular graphs.

Enumeration of unrooted orientable maps of arbitrary genus by number of edges and vertices

It is an outstanding open question in algorithmic graph theory to determine the complexity of the MAXIMUM INDEPENDENT SET problem on P_t-free graphs, that is, on graphs not containing any induced path on t vertices. So far, polynomial-time algorithms are known only for t at most 5 [Lokshtanov et al., SODA 2014, 570-581, 2014]. Here we study the existence of subexponential-time algorithms for the problem: by generalizing an earlier result of Randerath and Schiermeyer for t=5 [Discrete App. Math., 158 (2010), 1041-1044], we show that for any t at least 5, there is an algorithm for MAXIMUM INDEPENDENT SET on P_t-free graphs whose running time is subexponential in the number of vertices. SCATTERED SET is the generalization of MAXIMUM INDEPENDENT SET where the vertices of the solution are required to be at distance at least $d$ from each other. We give a complete characterization of those graphs H for which SCATTERED SET on H-free graphs can be solved in time subexponential in the size of the input (that is, in the number of vertices plus the number of edges): * If every component of H is a path, then d-SCATTERED SET on H-free graphs with n vertices and m edges can be solved in time 2^{(n+m)^{1-O(1/|V(H)|)}}, even if d is part of the input. * Otherwise, assuming ETH, there is no 2^{o(n+m)} time algorithm for d-SCATTERED SET for any fixed d at least 3 on H-free graphs with n vertices and m edges.

The crucial assumption of most of the model-based face analysis techniques is that any face can be generated by linear combinations of few faces. However, linear operations-Csuch as a simple pixelby- pixel addition of raw faces, are meaningless until pixel-wise correspondence of the 3D faces have been established. To overcome limitation of conventional mesh re-sampling algorithm, in this paper, we presented a fully automatic resolution, which can decrease the number of vertices and triangles while establishing correspondence between 3D faces. Based on facial features, a 3D face is segmented into 36 patches automatically at first. Each patch is then resampled to obtain a standardized patch, the same number of vertices with the template. At last, triangularly grid meshes are reconstructed for each patch individually and finally a standardized face is created by concatenating all the re-sampled patches. The experiment results show that without the complex iteration, our system is able to run quickly (resampling procedure costs less than one second) and obtain standardized 3D faces of different resolutions.

The complexity of a digraph property is the number of entries of the vertex adjacency matrix of a digraph which must be examined in worst case to determine whether the digraph has the property. Rivest and Vuillemin proved the result (conjectured by Aanderaa and Rosenberg) that every graph property that is monotone (preserved by addition of edges) and nontrivial (holds for some but not all graphs) has complexity θ(v2) where v is the number of vertices. Karp conjectured that every such property is evasive, i.e., requires that every entry of the incidence matrix be examined. In this paper it is shown that Karp's conjecture follows from another conjecture concerning group actions on topological spaces. A special case of this conjecture is proved and applied to prove Karp's conjecture for the case of properties of graph and digraph properties on a prime power number of vertices.

All Pairs Shortest Paths for Graphs with Small Integer Length Edges

Quasi-[formula omitted]-distance-balanced graphs

The various types of identical graphs having molecular structure of different degrees are isomorphic. In this paper, we study the structure of two graphs, which are isomorphic. Among the class of trees (star like tree) unicyclic, bicyclic, tricyclic, tetracyclic, pentacyclic, hexa-cyclic, heptacyclic etc. are isomorphic. Under discussion two graphs G_1 and G_2 are isomorphic because both graphs have same number of vertices and edges. We also point out different important applications of the various types of isomorphic graphs. In whole, we study about two graphs that can exist in different forms having the same number of vertices, edges having same degrees and the same edge connectivity. Such graphs called isomorphic graphs. While graph plotting and graph representation are sufficient topic in graph theory. In order to concentrate only on the abstract structure of graphs. A graph property defined to be a property to keep safe under all isomorphic of a graph. By brief look, we see that two graphs may appear very different when inspected visually, but they could have the same adjacency structure.