Approximating Average Parameters of Graphs

We study the power of four query models in the context of property testing in general graphs (i.e., with arbitrary edge densities), where our main case study is the problem of testing k-colorability. Two query types, which have been studied extensively in the past, are pair queries and neighbor queries. The former corresponds to asking whether there is an edge between any particular pair of vertices, and the latter to asking for the i’th neighbor of a particular vertex. We show that while for pair queries, testing k-colorability requires a number of queries that is a monotone decreasing function in the average degree d, the query complexity in the case of neighbor queries remains roughly the same for every density and for large values of k. We also consider a combined model that allows both types of queries, and we propose a new, stronger, query model, related to the field of Group Testing. We give one-sided error upper and lower bounds for all the models, where the bounds are nearly tight for three of the models. In some of the cases our lower bounds extend to two-sided error algorithms.The problem of testing k-colorability was previously studied in the contexts of dense and sparse graphs, and in our proofs we unify approaches from those cases, and also provide some new tools and techniques that may be of independent interest.Keywordspair queriesneighbor queriesgroup queriesk-colorability

In this paper we present a novel symmetric and dynamic access method, the Arc-tree, for organizing multidimensional data. The Arc-tree uses a new space-filling curve and a partition scheme that is based on bit interleaving. It divides the data space into non-overlapping arc partitions through splits imposed by meeting planes and co-centric spheres that alternate in a fixed order. The proposed structure is k-d-cut, fixed and brickwall. The Arc-tree inco rporates the properties of metric spaces and B-trees, is independent of data distribution and excludes the storage of empty partitions. Moreover, for a given data space the partitions are identical regardless of the order of insertions and deletions. The Arc-tree arranges partitions around a starting p oint and this makes the method especially suitable for applications where distance queries and searches on planes that intersect this point are concerned. We present the Arc-tree, describe its dynamic behavior and provide algorithms that prune the set of candidates to qualify partitions for several types of queries.

An approximate distance query data structure is a compact representation of a graph, and can be queried to approximate shortest paths between any pair of vertices. Any such data structure that retrieves stretch 2k Ω 1 paths must require space Ω(n1+1/k) for graphs of n nodes. The hard cases that enforce this lower bound are, however, rather dense graphs with average degree Ω(n1/k).

Improved routing strategies with succinct tables

A regular graph is a graph where each vertex has the same degree. A regular graph with vertices of degree k is called a k -regular graph or regular graph of degree k. Let G be a graph, the distance between two vertices in G is the number of edges in a shortest path connecting them. The diameter of G is the greatest distance between any pair of vertices. Let n4 be maximum integer number so that there exists an 4-regular graph on n4 vertices of diameter 2. We prove that n4 = 15.

The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.

For many technological networks, the network structures and the traffic taking place on them mutually interact. The demands of traffic increment spur the evolution and growth of the networks to maintain their normal and efficient functioning. In parallel, a change of the network structure leads to redistribution of the traffic. In this paper, we perform an extensive numerical and analytical study, extending results of Wang [Phys. Rev. Lett. 94, 188702 (2005)]. By introducing a general strength-coupling interaction driven by the traffic increment between any pair of vertices, our model generates networks of scale-free distributions of strength, weight, and degree. In particular, the obtained nonlinear correlation between vertex strength and degree, and the disassortative property demonstrate that the model is capable of characterizing weighted technological networks. Moreover, the generated graphs possess both dense clustering structures and an anticorrelation between vertex clustering and degree, which are widely observed in real-world networks. The corresponding theoretical predictions are well consistent with simulation results.

In this paper we consider the clustering coefficient, and clustering function in a random graph model proposed by Krioukov et al. in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most at a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, “short distances” and a non-vanishing clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $\alpha , \nu > 0$, which we think of as constant. Roughly speaking, the parameter $\alpha $ controls the power law exponent of the degree sequence and $\nu $ the average degree. Here we show that the clustering coefficient tends in probability to a constant $\gamma $ that we give explicitly as a closed form expression in terms of $\alpha , \nu $ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $\gamma (k)$ which we give explicitly as a closed form expression in terms of $\alpha , \nu $ and certain special functions. We are able to extend this last result also to sequences $(k_{n})_{n}$ where $k_{n}$ grows as a function of $n$. Our results show that $\gamma (k)$ scales differently, as $k$ grows, for different ranges of $\alpha $. More precisely, there exists constants $c_{\alpha ,\nu }$ depending on $\alpha $ and $\nu $, such that as $k \to \infty $, $\gamma (k) \sim c_{\alpha ,\nu } \cdot k^{2 - 4\alpha }$ if $\frac {1}{2} < \alpha < \frac {3}{4}$, $\gamma (k) \sim c_{\alpha ,\nu } \cdot \log (k) \cdot k^{-1}$ if $\alpha =\frac {3}{4}$ and $\gamma (k) \sim c_{\alpha ,\nu } \cdot k^{-1}$ when $\alpha > \frac {3}{4}$. These results contradict a claim of Krioukov et al., which stated that $\gamma (k)$ should always scale with $k^{-1}$ as we let $k$ grow.

The first and second Zagreb indices of a graph are equal‎, ‎respectively‎, ‎to‎ ‎the sum of squares of the vertex degrees‎, ‎and the sum of the products of‎ ‎the degrees of pairs of adjacent vertices‎. ‎We now consider analogous‎ ‎graph invariants‎, ‎based on the second degrees of vertices (number of‎ ‎their second neighbors)‎, ‎called leap Zagreb indices‎. ‎A number of their‎ ‎basic properties is established‎.

Clark proved that L(G) is hamiltonian if G is a connected graph of order n ≥ 6 such that deg u + deg v ≥ n – 1 – p(n) for every edge uv of G, where p(n) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n – 1 – p(n) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L(G) is pancyclic. Lesniak‐Foster and Williamson proved that G contains a spanning closed trail if |V(G)| = n ≥ 6, δ(G) ≥ 2 and deg u + deg v ≥ n – 1 for every pair of nonadjacent vertices u and v. The bound n – 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.

Let G be a simple graph of order n. Prime labeling is a bijective function f:V(G)→{1,2,…,n} such that gcd⁡(f(u),f(v))=1 for every pair of adjacent vertices u,v in G. A graph G that satisfies the definition of prime labeling is called a prime graph. The Dutch windmill graph D_r^n is a graph obtained by taking n copies of cycle graph C_r with a vertex in common. The double quadrilateral graph DQ is a graph constructed from two copies of C_4 and identifying one edge from each of them. The graph obtained by taking n copies of DQ and identifying one vertex of degree 3 from each of them as a common central vertex is called the double quadrilateral Dutch windmill graph DQ_n, for n≥1. Furthermore, graphs D_r^n and DQ_n becomes the base graph to construct two new graph classes, namely graph P_2 [D_r^n] and flower double quadrilateral graph FDQ_n. Both graph classes, constructed from the Dutch windmill graph, also contain even cycles. From previous research, it is known that graphs P_2 [D_4^n] and flower double quadrilateral graph FDQ_n have odd harmonious labeling. However, the determination of prime labeling on both classes is still an open problem. In this paper, we show that two classes of graphs constructed from Dutch windmill graphs with even cycles, namely graphs P_2 [D_4^n] and flower double quadrilateral graphs FDQ_n for n≥1 have a prime labeling. The result of this research shows that these graphs are prime graphs.

Triangle-free graphs whose independence number equals the degree

Due to the advance of the geo-spatial positioning and the computer graphics technology, digital terrain data become more and more popular nowadays. Query processing on terrain data has attracted considerable attention from both the academic community and the industry community. One fundamental and important query is the shortest distance query and many other applications such as proximity queries (including nearest neighbor queries and range queries), 3D object feature vector construction and 3D object data mining are built based on the result of the shortest distance query. In this paper, we study the shortest distance query which is to find the shortest distance between a point-of-interest and another point-of-interest on the surface of the terrain due to a variety of applications. As observed by existing studies, computing the exact shortest distance is very expensive. Some existing studies proposed e-approximate distance oracles where e is a non-negative real number and is an error parameter. However, the best-known algorithm has a large oracle construction time, a large oracle size and a large distance query time. Motivated by this, we propose a novel e-approximate distance oracle called the Space Efficient distance oracle (SE) which has a small oracle construction time, a small oracle size and a small distance query time due to its compactness storing concise information about pairwise distances between any two points-of-interest. Our experimental results show that the oracle construction time, the oracle size and the distance query time of SE are up to two orders of magnitude, up to 3 orders of magnitude and up to 5 orders of magnitude faster than the best-known algorithm.

Barabasi and Albert [1] find that many large networks exhibit a scale-free power-law distribution of vertex degrees. We show that when vertex degrees of large networks follow a scale-free power-law distribution with the exponent γ ≥ 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order less than log N. Furthermore, let nk be the number of degree-k vertices. In this paper we prove that n1 must be divisible by the least common multiple of kγ 1 , kγ 2 ,…, kγ l , where 1 = k1 kl is the degree sequence of the network. Then we construct a general model of networks of scale-free and obtain some detail properties on scale-free and small-world networks. Our method has the benefit of relying on conditions that are static and easily verified. They are verified by many experimental results of diverse real networks and have comprehensive applications to social, natural and synthetic systems.

By introducing and analyzing a renormalization procedure, Song et al. [1] draw the conclusion that many complex networks exhibit self-repeating patterns on all length scales. First, we aim to demonstrate that the aforementioned conclusion is inadequately justified, mainly because their equation (7) on the invariance of degree distribution under renormalization does not hold in general. Secondly, Barabási and Albert [2] find that many large networks exhibit a scale-free power-law distribution of vertex degrees. They show this common feature to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to those that are already well connected. We show that when vertex degrees of large networks follow a scale-free power-law distribution with the exponent γ ≥ 2, the number of degree-1 vertices, when nonzero, is of the same order as the network size N and that the average degree is of order less than log N. Given that many real networks satisfy these two conditions, our results add another necessary characteristic of the scale-free power-law distribution of vertex degrees in such networks. Our method has the benefit of relying on conditions that are static and easily verified. They are verified by many experimental results of diverse real networks.Keywordscomplex networkself-similarityscale-freecomputer networkcomputer communication