Maximum Vertex Degree Research Articles

Extremal Graphs to Vertex Degree Function Index for Convex Functions

The vertex-degree function index Hf(Γ) is defined as Hf(Γ)=∑v∈V(Γ)f(d(v)) for a function f(x) defined on non-negative real numbers. In this paper, we determine the extremal graphs with the maximum (minimum) vertex degree function index in the set of all n-vertex chemical trees, trees, and connected graphs. We also present the Nordhaus–Gaddum-type results for Hf(Γ)+Hf(Γ¯) and Hf(Γ)·Hf(Γ¯).

Groups with maximum vertex degree commuting graphs

Let G be a finite non-abelian group and Z(G) be its center. We associate a commuting graph \(\Gamma (G)\) to G, whose vertex set is \(G\setminus Z(G)\) and two distinct vertices are adjacent if they commute. In this paper we prove that the set of all non-abelian groups whose commuting graph has maximum vertex degree bounded above by a fixed \(k \in {\mathbb {N}}\) is finite. Also, we characterize all groups for which the associated commuting graphs have maximum vertex degree at most 4.

Bounds on Nirmala energy of graphs

Abstract The Nirmala matrix of a graph and its energy have recently defined. In this paper, we establish some features of the Nirmala eigenvalues. Then we propose various bounds on the Nirmala spectral radius and energy. Moreover, we derive a bound on the Nirmala energy including graph energy and maximum vertex degree.

Extremal trees of a given degree sequence or segment sequence with respect to average Steiner 3-eccentricity

The Steiner k-eccentricity of a vertex in a graph G is the maximum Steiner distance over all k-subsets containing the vertex. The average Steiner k-eccentricity of G is the mean value of all vertices’ Steiner k-eccentricities in G. Let Tn be the set of all n-vertex trees, Tn,Δ be the set of n-vertex trees with maximum degree Δ, Tn,Δk be the set of n-vertex trees with exactly k vertices of a given maximum degree Δ, and let MTnk be the set of n-vertex trees with exactly k vertices of maximum degree. In this paper, we first determine the sharp upper bound on the average Steiner 3-eccentricity of n-vertex trees with a given degree sequence. The corresponding extremal graphs are characterized. Consequently, together with majorization theory, all graphs among Tn,Δk (resp. Tn,Δ,MTnk,Tn) having the maximum average Steiner 3-eccentricity are identified. Then we characterize the unique n-vertex tree with a given segment sequence having the minimum average Steiner 3-eccentricity. Finally, we determine all n-vertex trees with a given number of segments having the minimum average Steiner 3-eccentricity.

Two-Step Colouring of Grid Graphs of Different Types

In this article, we consider the NP-hard problem of the two-step colouring of a graph. It is required to colour the graph in a given number of colours in a way, when no pair of vertices has the same colour, if these vertices are at a distance of 1 or 2 between each other. The optimum two-step colouring is one that uses the minimum possible number of colours.The two-step colouring problem is studied in application to grid graphs. We consider four types of grids: triangular, square, hexagonal, and octogonal. We show that the optimum two-step colouring of hexagonal and octogonal grid graphs requires 4 colours in the general case. We formulate the polynomial algorithms for such a colouring. A square grid graph with the maximum vertex degree equal to 3 requires 4 or 5 colours for a two-step colouring. In the paper, we suggest the backtracking algorithm for this case. Also, we present the algorithm, which works in linear time relative to the number of vertices, for the two-step colouring in 7 colours of a triangular grid graph and show that this colouring is always correct. If the maximum vertex degree equals 6, the solution is optimum.

A Characterization for the Neighbor-Distinguishing Index of Planar Graphs

Symmetry, such as structural symmetry, color symmetry and so on, plays an important role in graph coloring. In this paper, we use structural symmetry and color symmetry to study the characterization for the neighbor-distinguishing index of planar graphs. Let G be a simple graph with no isolated edges. The neighbor-distinguishing edge coloring of G is a proper edge coloring of G such that any two adjacent vertices admit different sets consisting of the colors of their incident edges. The neighbor-distinguishing index χa′(G) of G is the smallest number of colors in such an edge coloring of G. It was conjectured that if G is a connected graph with at least three vertices and G≠C5, then χa′(G)≤Δ+2. In this paper, we show that if G is a planar graph with maximum degree Δ≥13, then Δ≤χa′(G)≤Δ+1, and, further, χa′(G)=Δ+1 if and only if G contains two adjacent vertices of maximum degree.

Overfullness of critical class 2 graphs with a small core degree

Let G be a simple graph, and let n, Δ(G) and χ′(G) be the order, the maximum degree and the chromatic index of G, respectively. We call G overfull if |E(G)|/⌊n/2⌋>Δ(G), and critical if χ′(H)<χ′(G) for every proper subgraph H of G. Clearly, if G is overfull then χ′(G)=Δ(G)+1 by Vizing's Theorem. The core of G, denoted by GΔ, is the subgraph of G induced by all its maximum degree vertices. Hilton and Zhao conjectured that for any critical class 2 graph G with Δ(G)≥4, if the maximum degree of GΔ is at most two, then G is overfull, which in turn gives Δ(G)>n/2+1. We show that for any critical class 2 graph G, if the minimum degree of GΔ is at most two and Δ(G)>n/2+1, then G is overfull.

Universal Slope Sets for Upward Planar Drawings

We study universal sets of slopes for computing upward planar drawings of planar st-graphs. We first consider a subfamily of planar st-graphs, called bitonic st-graphs. We prove that every set mathcal {S} of varDelta slopes containing the horizontal slope is universal for 1-bend upward planar drawings of bitonic st-graphs with maximum vertex degree varDelta , i.e., every such digraph admits a 1-bend upward planar drawing whose edge segments use only slopes in mathcal {S}. This result is worst-case optimal in terms of number of slopes, and, for a suitable choice of mathcal {S}, it gives rise to drawings with worst-case optimal angular resolution. We then prove that every such set mathcal {S} can be used to construct 2-bend upward planar drawings of n-vertex planar st-graphs with at most 4n-9 bends in total.

Random Constructions of Hypergraphs with Large Girth and Without Panchromatic Colorings

We give probabilistic constructions of hypergraphs with large girth that do not admit panchromatic colorings. We prove new upper bounds on the minimal values of the number of edges and the maximum vertex degree in such hypergraphs.

On the Parameterized Complexity of the Expected Coverage Problem

The Maximum Covering Location Problem (MCLP) is a well-studied problem in the field of operations research. Given a network with positive or negative demands on the nodes, a positive integer k, the MCLP seeks to find k potential facility centers in the network such that the neighborhood coverage is maximized. We study the variant of MCLP where edges of the network are subject to random failures due to some disruptive events. One of the popular models capturing the unreliable nature of the facility location is the linear reliability ordering (LRO) model. In this model, with every edge e of the network, we associate its survival probability 0 ≤ pe ≤ 1, or equivalently, its failure probability 1 − pe. The failure correlation in LRO is the following: If an edge e fails then every edge e^{prime } with p_{e^{prime }} leq p_{e} surely fails. The task is to identify the positions of k facilities that maximize the expected coverage. We refer to this problem as Expected Coverage problem. We study the Expected Coverage problem from the parameterized complexity perspective and obtain the following results. 1. For the parameter pathwidth, we show that the Expected Coverage problem is W[1]-hard. We find this result a bit surprising, because the variant of the problem with non-negative demands is fixed-parameter tractable (FPT) parameterized by the treewidth of the input graph. 2. We complement the lower bound by the proof that Expected Coverage is FPT being parameterized by the treewidth and the maximum vertex degree. We give an algorithm that solves the problem in time 2^{{mathcal {O}}({textbf {tw}} log {varDelta })} n^{{mathcal {O}}(1)}, where tw is the treewidth, Δ is the maximum vertex degree, and n the number of vertices of the input graph. In particular, since Δ ≤ n, it means the problem is solvable in time n^{{mathcal {O}}({textbf {tw}})} , that is, is in XP parameterized by treewidth.

On Almost Self-centered Graphs and Almost Peripheral Graphs

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth of an almost self-centered graph of order $n$; (2) the maximum independence number of an almost self-centered graph of order $n$ and radius $r$; (3) the minimum order of a $k$-regular almost self-centered graph; (4) the maximum size of an almost peripheral graph of order $n$; (5) possible maximum degrees of an almost peripheral graph of order $n$ and (6) the maximum number of vertices of maximum degree in an almost peripheral graph of order $n$ with maximum degree $n-4$ which is the second largest possible. Whenever the extremal graphs have a neat form, we also describe them.

A Separator Theorem for Hypergraphs and a CSP-SAT Algorithm

We show that for every $r \ge 2$ there exists $\epsilon_r &gt; 0$ such that any $r$-uniform hypergraph with $m$ edges and maximum vertex degree $o(\sqrt{m})$ contains a set of at most $(\frac{1}{2} - \epsilon_r)m$ edges the removal of which breaks the hypergraph into connected components with at most $m/2$ edges. We use this to give an algorithm running in time $d^{(1 - \epsilon_r)m}$ that decides satisfiability of $m$-variable $(d, k)$-CSPs in which every variable appears in at most $r$ constraints, where $\epsilon_r$ depends only on $r$ and $k\in o(\sqrt{m})$. Furthermore our algorithm solves the corresponding #CSP-SAT and Max-CSP-SAT of these CSPs. We also show that CNF representations of unsatisfiable $(2, k)$-CSPs with variable frequency $r$ can be refuted in tree-like resolution in size $2^{(1 - \epsilon_r)m}$. Furthermore for Tseitin formulas on graphs with degree at most $k$ (which are $(2, k)$-CSPs) we give a deterministic algorithm finding such a refutation.

Inserting an edge into a geometric embedding

The algorithm to insert an edge e in linear time into a planar graph G with a minimal number of crossings on e[14] is a helpful tool for designing heuristics that minimize edge crossings in topological drawings of general graphs. Unfortunately, not all such topological drawings are stretchable, i.e., there may not exist an equivalent straight-line drawing. That is, there is no planar straight-line drawing Γ of G such that in Γ+e the edge e crosses the same edges as in the topological drawing of G+e and it does so in the same order. This motivates the study of the computational complexity of the problem Geometric Edge Insertion: Given a combinatorially embedded graph G, compute a geometric embedding Γ of G that minimizes the crossings in Γ+e.We give a characterization of the stretchable topological drawings of G+e that also applies to the case where the outer face is fixed; this answers an open question of Eades et al. [8]. Algorithmically, we focus on the case where the outer face is not fixed. We show that Geometric Edge Insertion can be solved efficiently for graphs of maximum degree 5. For the general case, we show a (Δ−2)-approximation, where Δ is the maximum vertex degree of G and an FPT algorithm with respect to the minimum number of crossings. Finally, we consider the problem of testing whether there exists a solution of Geometric Edge Insertion that achieves the lower bound obtained by a topological insertion.

Trees with extremal numbers of k-dominating sets

In this paper we investigate the relation between the number of k-dominating sets and the number of independent sets in a tree. We call the k-interior of a tree T the forest that contains all the vertices of T of degree at least k. Heuberger and Wagner (2008) characterized for all n,d≥3 the n-vertex trees Td,n with maximum vertex degree d and the maximum number of independent sets. We show that for all n,k≥2, if a tree T has the maximum number of k-dominating sets among all n-vertex trees, then either it contains exactly 2⌊n−2k−1⌋k-dominating sets or its k-interior is isomorphic to a forest aK1∪Tk,b for some integers a,b≥0. Moreover, we completely describe trees with the minimum possible number of k-dominating sets for every k≥2.

Extreme Wiener indices of trees with given number of vertices of maximum degree

In this paper, we consider trees of order n with a given number of vertices of maximum degree and solve the minimal extremal problem for the Wiener index on that class. Characteristics of the extremal structures are presented, some directly from previously established methods. We also consider the maximal value of the Wiener index in the class of trees of order n with exactly k vertices of a given maximum degree, and characterize appropriate optimal trees.

On the in–out–proper orientations of graphs

An orientation of a graph G is in–out–proper if any two adjacent vertices have different in–out-degrees, where the in–out-degree of each vertex is equal to the in-degree minus the out-degree of that vertex. The in–out–proper orientation number of a graph G, denoted by χ↔(G), is minD∈Γmaxv∈V(G)|dD±(v)|, where Γ is the set of in–out–proper orientations of G and dD±(v) is the in–out-degree of the vertex v in the orientation D. Borowiecki et al. proved that the in–out–proper orientation number is well-defined for any graph G (Borowiecki et al., 2012). So we have χ↔(G)≤Δ(G), where Δ(G) is the maximum degree of vertices in G. We conjecture that there exists a constant number c such that for every planar graph G, we have χ↔(G)≤c. Toward this speculation, we show that for every tree T we have χ↔(T)≤3 and this bound is sharp. Next, we study the in–out–proper orientation number of subcubic graphs. By using the properties of totally unimodular matrices we show that there is a polynomial time algorithm to determine whether χ↔(G)≤2, for a given graph G with maximum degree three. On the other hand, we show that it is NP-complete to decide whether χ↔(G)≤1 for a given bipartite graph G with maximum degree three. Finally, we study the in–out–proper orientation number of regular graphs.

On the maximum vertex degree of a conditional internet graph

On the maximum vertex degree of a conditional internet graph

Peg solitaire on graphs with large maximum degree

In 2011, Beeler and Hoilman introduced the peg solitaire on graphs. The peg solitaire on a connected graph is a one-player combinatorial game starting with exactly one hole in a vertex and pegs in all other vertices and removing all pegs but exactly one by a sequence of jumps; for a path [Formula: see text], if there are pegs in [Formula: see text] and [Formula: see text] and exists a hole in [Formula: see text], then [Formula: see text] can jump over [Formula: see text] into [Formula: see text], and after that, the peg in [Formula: see text] is removed. A problem of interest in the game is to characterize solvable (respectively, freely solvable) graphs, where a graph is solvable (respectively, freely solvable) if for some (respectively, any) vertex [Formula: see text], starting with a hole [Formula: see text], a terminal state consisting of a single peg can be obtained from the starting state by a sequence of jumps. In this paper, we consider the peg solitaire on graphs with large maximum degree. In particular, we show the necessary and sufficient condition for a graph with large maximum degree to be solvable in terms of the number of pendant vertices adjacent to a vertex of maximum degree. It is a notable point that this paper deals with a question of Beeler and Walvoort whether a non-solvable condition of trees can be extended to other graphs.

Aα -spectral radius and measures of graph irregularity

ABSTRACT Let G be a graph with order n and size m. For each real number , the -matrix of a graph G is defined as , where is the degree matrix of G, and is its adjacency matrix. The functions have been already used elsewhere as measures of graph irregularity. In this paper, we find several bounds for the spectral radius of the matrix . Especially, when G is non-empty we prove for that where and is the maximum vertex degree of G. These inequalities extend to the -matrix a Nikiforov's result involving the spectral radius of the adjacency matrix. In addition, for fixed order and size, we determine the graphs minimizing the degree variance and compute the -spectrum of some blown-up graphs.

On 3-Colouring Of Graphs with Short Faces and Bounded Maximum Vertex Degree

On 3-Colouring Of Graphs with Short Faces and Bounded Maximum Vertex Degree