Maximum Vertex Degree Research Articles

Размер памяти для хранения упорядоченного корневого графа

The paper considers boundaries of memory necessary and sufficient for storage of undirected ordered rooted connected graphs, both numbered and unnumbered. The introduction contains the basic definitions and the problem statement. A graph is rooted if one of its vertices is marked as a root. A graph is ordered if for each of its vertices all the incident edges are ordered (numbered). A graph is numbered if all its vertices are numbered with different integer numbers (from 0 to n -1, where n is the number of vertices). Two undirected ordered graphs G and G` are weakly isomorphic if there exists a one-to-one correspondence between their vertices, for which corresponding vertices has the same degrees (numbers of incident edges) and two edges having corresponding ends and the same numbers in these ends, also have the other ends corresponding. Isomorphism of rooted graphs should also correspond their roots. Isomorphism of numbered graphs should also correspond the vertices with the same numbers. Graphs are considered up to weak isomorphism. It is shown that the memory necessary and sufficient for storage of any graph has the size Q( mlogn ) for numbered graphs, Q( n +( m - n +1) logn ) for unnumbered graphs with the number of vertices n and the number of edges m , and Q( n 2 logn ) for graphs without multiple edges and loops with the number of vertices n . It is also shown that the memory sufficient for storage of an edge sequence of length O ( n ) or a spanning tree, has the O ( nlog ( n D)) or O ( nlog D) size, respectively, where D is the maximum vertex degree.

The general zeroth-order Randić index of maximal outerplanar graphs and trees with k maximum degree vertices

The general zeroth-order Randić index of maximal outerplanar graphs and trees with k maximum degree vertices

The height of faces of 3-polytopes

The height of a face in a 3-polytope is the maximum degree of the incident vertices of the face, and the height of a 3-polytope, h, is the minimum height of its faces. A face is pyramidal if it is either a 4-face incident with three 3-vertices, or a 3-face incident with two vertices of degree at most 4. If pyramidal faces are allowed, then h can be arbitrarily large; so we assume the absence of pyramidal faces. In 1940, Lebesgue proved that every quadrangulated 3-polytope has h ≤ 11. In 1995, this bound was lowered by Avgustinovich and Borodin to 10. Recently, we improved it to the sharp bound 8. For plane triangulation without 4-vertices, Borodin (1992), confirming the Kotzig conjecture of 1979, proved that h ≤ 20 which bound is sharp. Later, Borodin (1998) proved that h ≤ 20 for all triangulated 3-polytopes. Recently, we obtained the sharp bound 10 for triangle-free 3-polytopes. In 1996, Horňak and Jendrol’ proved for arbitrarily 3-polytopes that h ≤ 23. In this paper we improve this bound to the sharp bound 20.

On the bounds of the forgotten topological index

The forgotten topological index is defined as the sum of cubes of the degrees of the vertices of the molecular graph $G.$ In this paper, we obtain, analyze, and compare various lower bounds for the forgotten topological index involving the number of vertices, edges, and maximum and minimum vertex degree. Then we give Nordhaus--Gaddum-type inequalities for the forgotten topological index and coindex. Finally, we correct the number of extremal chemical trees on 15 vertices.

A Communication Game Related to the Sensitivity Conjecture

One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum degree of a real polynomial that interpolates the function) is bounded above by some fixed power of its sensitivity (which is the maximum vertex degree of the graph defined on the inputs where two inputs are adjacent if they differ in exactly one coordinate and their function values are different). We propose an attack on the sensitivity conjecture in terms of a novel two-player communication game. A lower bound of the form $n^{\Omega(1)}$ on the cost of this game would imply the sensitivity conjecture. To investigate the problem of bounding the cost of the game, three natural (stronger) variants of the question are considered. For two of these variants, protocols are presented that show that the hoped for lower bound does not hold. These protocols satisfy a certain monotonicity property, and (in contrast to the situation for the two variants) we show that the cost of any monotone protocol satisfies a strong lower bound. There is an easy upper bound of $\sqrt{n}$ on the cost of the game. We also improve slightly on this upper bound.

Bounds of graph energy in terms of vertex cover number

The energy E(G) of a graph G is the sum of the absolute values of all eigenvalues of G. In this paper, we give a lower bound and an upper bound for graph energy in terms of vertex cover number. For a graph G with vertex cover number τ, it is proved that 2τ−2c≤E(G)≤2τΔ, where c is the number of odd cycles in G and Δ is the maximum vertex degree of G. The lower bound is attained if and only if G is the disjoint union of some complete bipartite graphs with perfect matchings and some isolated vertices, the upper bound is attained if and only if G is the disjoint union of τ copies of K1,Δ together with some isolated vertices.

The graph of minimal distances of bent functions and its properties

A notion of the graph of minimal distances of bent functions is introduced. It is an undirected graph (V, E) where V is the set of all bent functions in 2k variables and $$(f, g) \in E$$ if the Hamming distance between f and g is equal to $$2^k$$ . It is shown that the maximum degree of the graph is equal to $$2^k (2^1 + 1) (2^2 + 1) \cdots (2^k + 1)$$ and all its vertices of maximum degree are quadratic bent functions. It is obtained that the degree of a vertex from Maiorana—McFarland class is not less than $$2^{2k + 1} - 2^k$$ . It is proven that the graph is connected for $$2k = 2, 4, 6$$ , disconnected for $$2k \ge 10$$ and its subgraph induced by all functions EA-equivalent to Maiorana—McFarland bent functions is connected.

On Universal Hypergraphs

A hypergraph $H$ is called universal for a family $\mathcal{F}$ of hypergraphs, if it contains every hypergraph $F \in \mathcal{F}$ as a copy. For the family of $r$-uniform hypergraphs with maximum vertex degree bounded by $\Delta$ and at most $n$ vertices any universal hypergraph has to contain $\Omega(n^{r-r/\Delta})$ many edges. We exploit constructions of Alon and Capalbo to obtain universal $r$-uniform hypergraphs with the optimal number of edges $O(n^{r-r/\Delta})$ when $r$ is even, $r \mid \Delta$ or $\Delta=2$. Further we generalize the result of Alon and Asodi about optimal universal graphs for the family of graphs with at most $m$ edges and no isolated vertices to hypergraphs.

Parameterized Complexity of Secluded Connectivity Problems

The Secluded Path problem models a situation where sensitive information has to be transmitted between a pair of nodes along a path in a network. The measure of the quality of a selected path is its exposure cost, which is the total cost of vertices in its closed neighborhood. The task is to select a secluded path, i.e., a path with a small exposure cost. Similarly, the Secluded Steiner Tree problem is to find a tree in a graph connecting a given set of terminals such that the exposure cost of the tree is minimized. In this paper we present a systematic study of the parameterized complexity of Secluded Steiner Tree. In particular, we establish the tractability of Secluded Path being parameterized by “above guarantee” value, which in this case is the length of a shortest path between vertices. We also show how to extend this result for Secluded Steiner Tree, in this case we parameterize above the size of an optimal Steiner tree and the number of terminals. We also consider various parameterization of the problems such as by the treewidth, the size of a vertex cover, feedback vertex set, or the maximum vertex degree and establish kernelization complexity of the problem subject to different choices of parameters.

AVD-total-chromatic number of some families of graphs with [formula omitted

An AVD-total-colouring of a simple graph G is a mapping π:V(G)∪E(G)→{1,…,k}, k≥1, such that: (i) for each pair of adjacent or incident elements x,y∈V(G)∪E(G), π(x)≠π(y); and (ii) for each pair of adjacent vertices x,y∈V(G), sets {π(x)}∪{π(xv):xv∈E(G),v∈V(G)} and {π(y)}∪{π(yv):yv∈E(G),v∈V(G)} are distinct. The AVD-total-chromatic number, χa″(G), is the smallest number of colours for which G admits an AVD-total-colouring. In 2010, J. Hulgan conjectured that any simple graph G with maximum degree three has χa″(G)≤5. In this article, we verify Hulgan’s Conjecture for simple graphs G with Δ(G)=3 and without adjacent vertices of maximum degree, and also for the following families of snarks: the flower snarks, generalized Blanuša snarks, and LP1-snarks. In fact, we determine the exact value of χa″(G) for all families considered in this work.

Spanning structures and universality in sparse hypergraphs

AbstractIn this paper the problem of finding various spanning structures in random hypergraphs is studied. We notice that a general result of Riordan [Combin Probab Comput 9 (2000), 125–148] can be adapted from random graphs to random r‐uniform hypergraphs and provide sufficient conditions when a random r‐uniform hypergraph contains a given spanning structure a.a.s. We also discuss several spanning structures such as cube‐hypergraphs, lattices, spheres, and Hamilton cycles in hypergraphs. Moreover, we study universality, i.e. when does an r‐uniform hypergraph contain any hypergraph on n vertices and with maximum vertex degree bounded by Δ? For it is shown that this holds for a.a.s. by combining approaches taken by Dellamonica, Kohayakawa, Rödl and Ruciński [Random Struct Algorithms 46 (2015), 274–299] and of Ferber, Nenadov and Peter [Random Struct Algorithms 48 (2016), 546–564] and of Kim and Lee [SIAM J Discrete Math 28 (2014), 1467–1478]. Furthermore it is shown that the random graph G(n, p) for appropriate p and explicit constructions of universal graphs due to Alon, Capalbo, Kohayakawa, Rödl, Ruciński and Szemerédi [Lecture Notes in Comput. Sci., Vol. 2129, Springer, Berlin, 2001, pp. 170–180] and Alon and Capalbo [Random Struct Algorithms 31 (2007), 123–133; Proceedings of the 9th Annual ACM‐SIAM Symposium Society of Industrial and Applied Mathematics, 2008, pp. 373–378] yield constructions of universal hypergraphs that are sparser than the random hypergraph with © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 49, 819–844, 2016

On conditional configuration graphs with random distribution of vertex degrees

We consider a configuration graph with N vertices. The degrees of the vertices are drawn independently from a discrete power-law distribution with positive parameter τ . They are equal to the number of each vertex’s numbered semiedges. The graph is constructed by joining all of the semiedges pairwise equiprobably to form edges. Research in the last years showed that configuration power-law random graphs with τ ∈ (1, 2) are deemed to be a good implementation of Internet topology. Such graphs could be used also for modeling forest fires as well as banking system defaults. But in these cases usually τ > 2. Parameter τ may depend on N and even be random. In the paper we consider configuration random graphs under the condition that the sum of vertex degrees is equal to n. Random graph dynamics as N → ∞ is assumed to take place in a random environment, where τ is a random variable following uniform distribution on the interval [a, b], 0 < a < b < ∞. We obtained the limit distributions of the maximum vertex degree and the number of vertices with a given degree as N, n → ∞.

Distributed Greedy Approximation to Maximum Weighted Independent Set for Scheduling With Fading Channels

It has been known that scheduling algorithms designed to achieve throughput optimality and good delay performance often require solving the Maximum Weighted Independent Set (MWIS) problem. However, under most realistic network settings, the MWIS problem is known to be NP-hard. In non-fading environments, low-complexity scheduling algorithms have been provided that converge either to the MWIS solution in time or to a solution that achieves at least a provable fraction of the achievable throughput. However, in more practical systems the channel conditions can vary at faster time-scales than convergence occurs in these lower-complexity algorithms. Hence, these algorithms cannot take advantage of opportunistic gains, and may no longer result in achieving good performance. In this paper, we propose a low-complexity scheduling scheme that performs provably well under fading channels and is amenable to implement in a distributed manner. To the best of our knowledge, this is the first scheduling scheme under fading environments that requires only local information, has a low complexity that grows logarithmically with the network size (provided that the conflict graph has bounded maximum vertex degree), and achieves provable performance guarantees (arbitrarily close to that of the well-known centralized Greedy Maximal Scheduler). We verify that the throughput and the delay of our proposed scheme are close to those of the optimal MaxWeight that solves MWIS at each time. Further, we implement our algorithm in a testbed by modifying the existing IEEE 802.11 DCF. The experiment results show that our implementation successfully accounts for wireless fading, attains the short-term opportunistic gains in practice, and hence substantially outperforms IEEE 802.11 DCF.

Adjacent vertex distinguishing total colorings of 2-degenerate graphs

Let ϕ be a proper total coloring of G. We use Cϕ(v)={ϕ(v)}∪{ϕ(uv)∣uv∈E(G)} to denote the set of colors assigned to a vertex v and those edges incident with v. An adjacent vertex distinguishing total coloring of a graph G is a proper total coloring of G such that Cϕ(u)≠Cϕ(v) for any uv∈E(G). The minimum number of colors required for an adjacent vertex distinguishing total coloring of G is denoted by χa″(G). In this paper we show that if G is a 2-degenerate graph, then χa″(G)≤max{Δ(G)+2,6}. Moreover, we also show that when Δ≥5, χa″(G)=Δ(G)+2 if and only if G contains two adjacent vertices of maximum degree. Our results imply the results on outerplanar graphs (Wang and Wang, 2010), K4-minor free graphs (Wang and Wang, 2009) and graphs with maximum average degree less than 3 (Wang and Wang, 2008).

A duality transform for constructing small grid embeddings of 3d polytopes

We study the problem of how to obtain an integer realization of a 3d polytope when an integer realization of its dual polytope is given. We focus on grid embeddings with small coordinates and develop novel techniques based on Colin de Verdière matrices and the Maxwell–Cremona lifting method.We show that every truncated 3d polytope with n vertices can be realized on a grid of size O(n9log⁡6+1). Moreover, for every simplicial 3d polytope with n vertices with maximal vertex degree Δ and vertices placed on an L×L×L grid, a dual polytope can be realized on an integer grid of size O(nL3Δ+9). This implies that for a class C of simplicial 3d polytopes with bounded vertex degree and polynomial size grid embedding, the dual polytopes of C can be realized on a polynomial size grid as well.

Some new formulae on Δ–optimum exclusive sum labeling of certain trees

All sum graphs are disconnected. The notions of sum labeling and sum graphs were introduced by Harary [1] and later extended to include all integers [2]. A mapping L is called a sum labeling a graph H(V (H), E(H)) if it is an injection from V(H) to a set of positive integers such that xy ∈E(H) if and only if there exists a vertex w ∈V(H) such that L(w)=L(x) + L(y). In this case w is called a working vertex. We call L as an exclusive sum labeling of a graph G if it is sum labeling of for some non negative integer r and G contains no working vertex. In general, a graph G will require some isolated vertices to be labeled exclusively. The least possible number of such isolated vertices is called exclusive sum number of G, denoted by ∈(G). An exclusive sum labeling of a graph G is said to be optimum if it labels G exclusively by using ∈(G) isolated vertices. In case ∈(G)=Δ(G), where D(G) denotes the maximum degree of vertices in G, the labeling is called Δ- optimum exclusive sum labeling. In this paper we developed some new formulae to find Δ- optimum exclusive sum labeling of certain trees.

On the limit distributions of the degrees of vertices in configuration graphs with a bounded number of edges

A model of a configuration graph on vertices is considered in which the degrees of the vertices are distributed identically and independently according to the law , , , and the number of edges is no greater than . Limit theorems for the number of vertices of a particular degree and for the maximum vertex degree as are established. Bibliography: 18 titles.

Trees with given maximum degree minimizing the spectral radius

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

Graphs with the Strong Havel–Hakimi Property

The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel-Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel-Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.

Minimal k-Connected Graphs with Minimal Number of Vertices of Degree k

A graph is k-connected if it has at least k+1 vertices and remains connected after deleting any k−1 vertices. A k-connected graph is said to be minimal if any its subgraph obtained by deleting any edge is not k-connected. W. Mader proved that any minimal k-connected graph with n vertices has at least $$ \frac{\left(k-1\right)n+2k}{2k-1} $$ vertices of degree k. The main result of the present paper is that any minimal k-connected graph with minimal number of vertices of degree k is isomorphic to a graph G k,T , where T is a tree the maximal vertex degree of which is at most k + 1. The graph G k,T is constructed from k disjoint copies of the tree T in the following way. If a is a vertex of T of degree j and a 1 , . . . , a k are the corresponding vertices of the copies of T, then k + 1 − j new vertices of degree k, which are adjacent to {a 1 , . . . , a k }, are added. Bibliography: 10 titles.