Maximum Vertex Degree Research Articles

An efficient case for computing minimum linear arboricity with small maximum degree

Graph coloring has interesting applications in optimization, calculation of Hessian matrix, network design and so on. In this paper, we consider an improper edge coloring which is one important coloring-linear arboricity. For a graph G, a linear forest is a disjoint union of paths and cycles. The linear arboricity la(G) is the minimum number of disjoint linear forests such that their union is exactly the edge set of G. In this paper, we study a special case that G is a simple planar graph with two not adjacent cycles each with a chordal and length between 4 and 7. We show that in this special case, $$la(G)=\lceil \frac{\Delta }{2}\rceil $$ where $$\Delta $$ is the maximum vertex degree of G and $$\Delta \ge 7$$ .

New bounds on the hyper-Zagreb index for the simple connected graphs

The hyper-Zagreb index of a simple connected graph G is defined by ${\chi ^2}(G) = \sum_{uv \in E(G)} {{{\left( {d(u) + d(v)} \right)}^2}}$. In this paper, we establish, analyze and compare some new upper bounds on the Hyper-Zagreb index in terms of the number of vertices n, number of edges m, maximum vertex degree $\Delta$, and minimum vertex degree $\delta$, first Zagreb index M_1(G), second Zagreb index M_2(G), harmonic index H(G), and inverse edge degree IED(G). In addition, we give the identities on Hyper-Zagreb index and its coindex for the simple connected graphs.

Optimal channel assignment and L(p, 1)-labeling

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function $$f:V(G)\rightarrow \{0,1,2,\ldots ,k\}$$ such that $$|f(u)-f(v)|\ge p$$ if $$d(u,v)=1$$ and $$|f(u)-f(v)|\ge 1$$ if $$d(u,v)=2$$ , where d(u, v) is the distance between the two vertices u and v in the graph. Denote $$\lambda _{p,1}^l(G)= \min \{k \mid G$$ has a list k-L(p, 1)-labeling $$\}$$ . In this paper we show upper bounds $$\lambda _{1,1}^l(G)\le \Delta +9$$ and $$\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}$$ for planar graphs G without 4- and 6-cycles, where $$\Delta $$ is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.

Bipartite graphs with bounded vertex degree and largest eigenvalue of the form [formula omitted

In this note, we generalize two results from [2] and [3] and for every Δ and Δ ≤ r ≤ Δ2 construct a family of bipartite graphs with the maximal vertex degree Δ and largest eigenvalue equal to r.

Bakry–Émery curvature and diameter bounds on graphs

We prove diameter bounds for graphs having positive Ricci-curvature bound in Bakry-Emery sense. One result using only curvature and maximal vertex degree is sharp in case of hypercubes. The other result depends on an additional dimension bound, but is independent of the vertex degree. In particular, the second result is the first Bonnet-Myers type theorem for unbounded graph Laplacians. Moreover, our results improve diameter bounds from [6] and [10].

Total coloring of rooted path graphs

A total coloring of a graph is an assignment of colors to both its vertices and edges so that adjacent or incident elements acquire distinct colors. In this note, we give a simple greedy algorithm to totally color a rooted path graph G with at most Δ(G)+2 colors, where Δ(G) is the maximum vertex degree of G. Our algorithm is inspired by a method by Bojarshinov (2001) [3] for interval graphs and provides a new proof that the Total Coloring Conjecture, posed independently by Behzad (1965) [1] and Vizing (1968) [15], holds for rooted path graphs. In the process, we also prove a useful property of greedy neighborhood coloring for chordal graphs.

The distance spectral radius of trees

The unique graphs with maximum distance spectral radius among trees with given number of vertices of maximum degree and among homeomorphically irreducible trees, respectively, are determined.

New results on edge partitions of 1-plane graphs

A 1-plane graph is a graph embedded in the plane such that each edge is crossed at most once. A NIC-plane graph is a 1-plane graph such that any two pairs of crossing edges share at most one end-vertex. An edge partition of a 1-plane graph G is a coloring of the edges of G with two colors, red and blue, such that both the graph induced by the red edges and the graph induced by the blue edges are plane graphs. We prove the following: (i) Every NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree three; this bound on the vertex degree is worst-case optimal. (ii) Deciding whether a NIC-plane graph admits an edge partition such that the red graph has maximum vertex degree two is NP-complete. (iii) Deciding whether a 1-plane graph admits an edge partition such that the red graph has maximum vertex degree one, and computing one in the positive case, can be done in quadratic time. Applications of these results to graph drawing are also discussed.

Generic Local Hamiltonians are Gapless

We prove that generic quantum local Hamiltonians are gapless. In fact, we prove that there is a continuous density of states above the ground state. The Hamiltonian can be on a lattice in any spatial dimension or on a graph with a bounded maximum vertex degree. The type of interactions allowed for include translational invariance in a disorder (i.e., probabilistic) sense with some assumptions on the local distributions. Examples include many-body localization and random spin models. We calculate the scaling of the gap with the system's size when the local terms are distributed according to a Gaussian β orthogonal random matrix ensemble. As a corollary, there exist finite size partitions with respect to which the ground state is arbitrarily close to a product state. When the local eigenvalue distribution is discrete, in addition to the lack of an energy gap in the limit, we prove that the ground state has finite size degeneracies. The proofs are simple and constructive. This work excludes the important class of truly translationally invariant Hamiltonians where the local terms are all equal.

On limit behavior of maximum vertex degree in a conditional configuration graph near critical points

Abstract We consider configuration graphs with N vertices. The degrees of vertices are independent identically distributed random variables having the power-law distribution with parameter τ > 0. There are two critical values of this parameter: τ = 1 and τ = 2. The properties of a graph change significantly when τ = τ(N) passes these points as N → ∞. Let G N,n be the subset of random graphs under the condition that sum of degrees of its vertices is equal to n. The limit theorem for the maximum vertex degree in G N,n as N,n → ∞ and τ → 1 or τ → 2 is proved.

Controllable Graphs with Small Sums of Diameters and Maximum Vertex Degrees

A novel graph structure modeling a linear time invariant system following the Laplacian dynamics is proposed. This structure is based on the combination of two graphs and each of them has exactly two vertices with the same degree. It is shown that if the difference of vertex numbers of the two graphs is at most one then the combined graph is controllable by a single controller. A sufficient condition concerning the vertex selection for receiving control signals to guarantee the controllability of the combined graph is also proposed. This structure features a small sum of the maximum vertex degree and the diameter compared to some well-known controllable graphs. Numerical examples are provided to illustrate our work.

Configurations containing a given linear hypergraph

AbstractWe investigate the minimum order of a linear r‐regular k‐uniform hypergraph, also known as an ‐combinatorial configuration, which contains a given linear k‐uniform hypergraph of maximum (vertex) degree at most r.

L(0, 1)-Labelling of Trapezoid Graphs

L(0, 1)-labelling of a graph $$G=(V,E)$$ is a function f from the vertex set V(G) to the set of non-negative integers such that adjacent vertices get number zero apart, and vertices at distance two get distinct numbers. The L(0, 1)-labelling number denoted by $$\lambda _{0,1}(G)$$ of G is the minimum range of labels over all such labelling. In this article, it is shown that, for a trapezoid graph G with maximum vertex degree $$\Delta $$ , the upper bound of $$\lambda _{0,1}(G)$$ is $$\Delta -1$$ .

Parameterized complexity of team formation in social networks

Given a task that requires some skills and a social network of individuals with different skills, the Team Formation problem asks to find a team of individuals that together can perform the task, while minimizing communication costs. Since the problem is NP-hard, we identify the source of intractability by analyzing its parameterized complexity with respect to parameters such as the total number of skills k, the team size l, the communication cost budget b, and the maximum vertex degree Δ. We show that the computational complexity strongly depends on the communication cost measure: when using the weight of a minimum spanning tree of the subgraph formed by the selected team, we obtain fixed-parameter tractability for example with respect to the parameter k. In contrast, when using the diameter as measure, the problem is intractable with respect to any single parameter; however, combining Δ with either b or l yields fixed-parameter tractability.

On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree

The first multiplicative Zagreb index of a graph G is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of ∏1(G) and ∏2(G) of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.

Random Graphs' Robustness in Random Environment

We consider configuration graphs the vertex degrees of which are independent and follow the power-law distribution. Random graphs dynamics takes place in a random environment with the parameter of vertex degree distribution following uniform distributions on finite fixed intervals. As the number of vertices tends to infinity the limit distributions of the maximum vertex degree and the number of vertices with a given degree were obtained. By computer simulations we study the robustness of those graphs from the viewpoints of link saving and node survival in the two cases of the destruction process: the ``targeted attack'' and the ``random breakdown''. We obtained and compared the results under the conditions that the vertex degree distribution was averaged with respect to the distribution of the power-law parameter or that the values of the parameter were drawn from the uniform distribution separately for each vertex.

Improved Approximation Algorithms for the Maximum Happy Vertices and Edges Problems

The Maximum Happy Vertices (MHV) problem and the Maximum Happy Edges (MHE) problem are two fundamental problems arising in the study of the homophyly phenomenon in large scale networks. Both of these two problems are NP-hard. Interestingly, the MHE problem is a natural generalization of Multiway Uncut, the complement of the classic Multiway Cut problem. In this paper, we present new approximation algorithms for MHV and MHE based on randomized LP-rounding techniques. Specifically, we show that MHV can be approximated within \(\frac{1}{\varDelta +1}\), where \({\varDelta }\) is the maximum vertex degree, and MHE can be approximated within \(\frac{1}{2} + \frac{\sqrt{2}}{4}f(k) \ge 0.8535\), where \(f(k) \ge 1\) is a function of the color number k. These results improve on the previous approximation ratios for MHV, MHE as well as Multiway Uncut in the literature.

Bounds for the augmented Zagreb and the atom-bond connectivity indices

Using electrical networks and majorization we obtain a lower bound for the augmented Zagreb index in terms of the number of vertices and edges, and the maximum vertex degree. An analogous upper bound for the atom-bond connectivity index is also obtained. These bounds, which are attained by the complete graph Kn, are not comparable to the ones found in the literature.

On Adjacent Vertex-distinguishing Total Chromatic Number of Generalized Mycielski Graphs

The adjacent vertex-distinguishing total chromatic number of a graph $G$, denoted by $\chi_{at}(G)$, is the smallest $k$ for which $G$ has a proper total $k$-coloring such that any two adjacent vertices have distinct sets of colors appearing on the vertex and its incident edges. In regard of this number, there is a famous conjecture (AVDTCC) which states that for any simple graph $G$, $\chi_{at}(G) \leq \Delta(G)+3$. In this paper, we study this number for the generalized Mycielski graph $\mu_m(G)$ of a graph $G$. We prove that the satisfiability of the conjecture AVDTCC in $G$ implies its satisfiability in $\mu_m(G)$. Particularly we give the exact values of $\chi_{at}(\mu_m(G))$ when $G$ is a graph with maximum degree less than $3$ or a complete graph. Moreover, we investigate $\chi_{at}(G)$ for any graph $G$ with only one maximum degree vertex by showing that $\chi_{at}(G) \leq \Delta(G)+2$ when $\Delta(G) \leq 4$.

Vertex-Coloring with Defects

Defective coloring is a variant of the traditional vertex-coloring in which adjacent vertices are allowed to have the same color, as long as the induced monochromatic components have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, diameter, and acyclicity of the monochromatic components. We focus on defective colorings with $\kappa$ colors in which the monochromatic components are acyclic and have small diameter, namely we consider (edge, $\kappa$)-colorings, in which the monochromatic components have diameter $1$, and (star, $\kappa$)-colorings, in which they have diameter $2$. We prove that the (edge, $3$)-coloring problem remains NP-complete even for graphs with maximum vertex-degree $6$, hence answering an open question posed by Cowen et al. [Cowen, Goddard, Jesurum, J. Graph Theory, 1997], and for planar graphs with maximum vertex-degree $7$, and we prove that the (star, $3$)-coloring problem is NP-complete even for planar graphs with bounded maximum vertex-degree. On the other hand, we give linear-time algorithms for testing the existence of (edge, $2$)-colorings and of (star, $2$)-colorings of partial $2$-trees. Finally, we prove that outerpaths, a notable subclass of outerplanar graphs, always admit (star, $2$)-colorings.