L(2,1)-Edge-Labelings of the Edge-Path-Replacement of a Graph

2018 ◽  
Vol 29 (01) ◽  
pp. 91-100
Author(s):  
Nianfeng Lin ◽  
Damei Lü ◽  
Jinhua Wang
Keyword(s):  
Maximum Degree ◽  
Common Edge ◽  
Edge Path ◽  
The Difference ◽  
Edge Labeling

Two edges [Formula: see text] and [Formula: see text] in a graph [Formula: see text] are said to be adjacent if they have a vertex in common, and distance two apart if they are nonadjacent but both are adjacent to a common edge. An [Formula: see text]-edge-labeling of a graph [Formula: see text] is an assignment of nonnegative integers, called labels, to the edges of [Formula: see text] such that the difference between labels of any two adjacent edges is at least [Formula: see text], and the labels of any two edges that are distance two apart are different. The span of an [Formula: see text]-edge-labeling of a graph [Formula: see text] is the difference between the maximum and minimum labels. The minimum span over all [Formula: see text]-edge-labelings of a graph [Formula: see text] is called the [Formula: see text]-edge-labeling number of [Formula: see text], denoted by [Formula: see text]. For [Formula: see text], the edge-path-replacement of a graph [Formula: see text], denoted by [Formula: see text], is a graph obtained by replacing each edge of [Formula: see text] with a path [Formula: see text] on [Formula: see text] vertices. This paper investigates the [Formula: see text]-edge-labeling number of the edge-path-replacement [Formula: see text] of a graph [Formula: see text] for [Formula: see text]. We get the following main results: [Formula: see text] Let [Formula: see text] be a graph with maximum degree [Formula: see text] and [Formula: see text] be an integer not less than [Formula: see text], then [Formula: see text] if [Formula: see text] is odd, and otherwise [Formula: see text]. [Formula: see text] Let [Formula: see text] be a graph with maximum degree [Formula: see text]. Then [Formula: see text] when [Formula: see text] is even, and [Formula: see text] when [Formula: see text] is odd.

scholarly journals List-antimagic labeling of vertex-weighted graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Zhanar Berikkyzy ◽  
Axel Brandt ◽  
Sogol Jahanbekam ◽  
Victor Larsen ◽  
Danny Rorabaugh
Keyword(s):  
Oriented Graph ◽  
Weighted Graphs ◽  
Antimagic Labeling ◽  
The Difference ◽  
Edge Labeling

A graph $G$ is $k$-$weighted-list-antimagic$ if for any vertex weighting $\omega\colon V(G)\to\mathbb{R}$ and any list assignment $L\colon E(G)\to2^{\mathbb{R}}$ with $|L(e)|\geq |E(G)|+k$ there exists an edge labeling $f$ such that $f(e)\in L(e)$ for all $e\in E(G)$, labels of edges are pairwise distinct, and the sum of the labels on edges incident to a vertex plus the weight of that vertex is distinct from the sum at every other vertex. In this paper we prove that every graph on $n$ vertices having no $K_1$ or $K_2$ component is $\lfloor{\frac{4n}{3}}\rfloor$-weighted-list-antimagic. An oriented graph $G$ is $k$-$oriented-antimagic$ if there exists an injective edge labeling from $E(G)$ into $\{1,\dotsc,|E(G)|+k\}$ such that the sum of the labels on edges incident to and oriented toward a vertex minus the sum of the labels on edges incident to and oriented away from that vertex is distinct from the difference of sums at every other vertex. We prove that every graph on $n$ vertices with no $K_1$ component admits an orientation that is $\lfloor{\frac{2n}{3}}\rfloor$-oriented-antimagic.

scholarly journals On Bounding the Difference of the Maximum Degree and the Clique Number

2014 ◽  
Vol 31 (5) ◽  
pp. 1689-1702 ◽  
Author(s):  
Oliver Schaudt ◽  
Vera Weil
Keyword(s):  
Maximum Degree ◽  
Clique Number ◽  
The Difference

Total Labelling of the Tree

2011 ◽  
Vol 105-107 ◽  
pp. 2275-2278 ◽  
Author(s):  
Hai Na Sun
Keyword(s):  
Maximum Degree ◽  
The Difference

. The (2,1)-total labelling of the tree has been widely studied. In this paper we study the (3,1)-total labelling number of the tree. The (3,1)-total labelling number of the tree is the width of the smallest range of integers to label the vertices and the edges such that no two adjacent vertices or two adjacent edges have the same labels and the difference between the labels of a vertex and its incident edges is at least 3. We prove that if the distance of the maximum degree in the tree is not 2, then the (3,1)-total labelling number is the maximum degree plus 3.

scholarly journals An exact algorithm for the generalized list T-coloring problem

2014 ◽  
Vol Vol. 16 no. 3 (Discrete Algorithms) ◽  
Author(s):  
Konstanty Junosza-Szaniawski ◽  
Pawel Rzazewski
Keyword(s):  
Graph Coloring ◽  
Channel Assignment ◽  
Maximum Degree ◽  
Perfect Matching ◽  
Exact Algorithm ◽  
Special Structure ◽  
Input Graph ◽  
Discrete Algorithms ◽  
International Audience ◽  
The Difference

Discrete Algorithms International audience The generalized list T-coloring is a common generalization of many graph coloring models, including classical coloring, L(p,q)-labeling, channel assignment and T-coloring. Every vertex from the input graph has a list of permitted labels. Moreover, every edge has a set of forbidden differences. We ask for a labeling of vertices of the input graph with natural numbers, in which every vertex gets a label from its list of permitted labels and the difference of labels of the endpoints of each edge does not belong to the set of forbidden differences of this edge. In this paper we present an exact algorithm solving this problem, running in time O*((τ+2)n), where τ is the maximum forbidden difference over all edges of the input graph and n is the number of its vertices. Moreover, we show how to improve this bound if the input graph has some special structure, e.g. a bounded maximum degree, no big induced stars or a perfect matching.

scholarly journals On the complexity of the balanced vertex ordering problem

2007 ◽  
Vol Vol. 9 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Jan Kára ◽  
Jan Kratochvil ◽  
David R. Wood
Keyword(s):  
Polynomial Time ◽  
Maximum Degree ◽  
Graph Drawing ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Np Hard ◽  
Vertex Ordering ◽  
The Difference ◽  
Balanced Ordering ◽  
Applied Math

Graphs and Algorithms International audience We consider the problem of finding a balanced ordering of the vertices of a graph. More precisely, we want to minimise the sum, taken over all vertices v, of the difference between the number of neighbours to the left and right of v. This problem, which has applications in graph drawing, was recently introduced by Biedl et al. [Discrete Applied Math. 148:27―48, 2005]. They proved that the problem is solvable in polynomial time for graphs with maximum degree three, but NP-hard for graphs with maximum degree six. One of our main results is to close the gap in these results, by proving NP-hardness for graphs with maximum degree four. Furthermore, we prove that the problem remains NP-hard for planar graphs with maximum degree four and for 5-regular graphs. On the other hand, we introduce a polynomial time algorithm that determines whetherthere is a vertex ordering with total imbalance smaller than a fixed constant, and a polynomial time algorithm that determines whether a given multigraph with even degrees has an 'almost balanced' ordering.

scholarly journals Difference betwwen the maximum degree and the index of a graph

2019 ◽  
Vol 54 (4) ◽  
pp. 434-440
Author(s):  
V. I. Benediktovich
Keyword(s):  
Lower Bound ◽  
Spectral Radius ◽  
Lower Bounds ◽  
Maximum Degree ◽  
Lower Boundary ◽  
Arbitrary Graph ◽  
Analogous Statement ◽  
Graph Classes ◽  
The Difference ◽  
A Chain

An algebraic parameter of a graph – a difference between its maximum degree and its spectral radius is considered in this paper. It is well known that this graph parameter is always nonnegative and represents some measure of deviation of a graph from its regularity. In the last two decades, many papers have been devoted to the study of this parameter. In particular, its lower bound depending on the graph order and diameter was obtained in 2007 by mathematician S. M. Cioabă. In 2017 when studying the upper and the lower bounds of this parameter, M. R. Oboudi made a conjecture that the lower bound of a given parameter for an arbitrary graph is the difference between a maximum degree and a spectral radius of a chain. This is very similar to the analogous statement for the spectral radius of an arbitrary graph whose lower boundary is also the spectral radius of a chain. In this paper, the above conjecture is confirmed for some graph classes.

scholarly journals Transversals and Independence in Linear Hypergraphs with Maximum Degree Two

10.37236/6160 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Michael A. Henning ◽  
Anders Yeo
Keyword(s):  
Maximum Degree ◽  
Independence Number ◽  
Independent Set ◽  
Upper Bounds ◽  
Maximum Cardinality ◽  
Common Edge ◽  
Minimum Cardinality ◽  
Uniform Hypergraphs ◽  
Transversal Number ◽  
Linear Hypergraphs

For $k \ge 2$, let $H$ be a $k$-uniform hypergraph on $n$ vertices and $m$ edges. Let $S$ be a set of vertices in a hypergraph $H$. The set $S$ is a transversal if $S$ intersects every edge of $H$, while the set $S$ is strongly independent if no two vertices in $S$ belong to a common edge. The transversal number, $\tau(H)$, of $H$ is the minimum cardinality of a transversal in $H$, and the strong independence number of $H$, $\alpha(H)$, is the maximum cardinality of a strongly independent set in $H$. The hypergraph $H$ is linear if every two distinct edges of $H$ intersect in at most one vertex. Let $\mathcal{H}_k$ be the class of all connected, linear, $k$-uniform hypergraphs with maximum degree $2$. It is known [European J. Combin. 36 (2014), 231–236] that if $H \in \mathcal{H}_k$, then $(k+1)\tau(H) \le n+m$, and there are only two hypergraphs that achieve equality in the bound. In this paper, we prove a much more powerful result, and establish tight upper bounds on $\tau(H)$ and tight lower bounds on $\alpha(H)$ that are achieved for  infinite families of hypergraphs. More precisely, if $k \ge 3$ is odd and $H \in \mathcal{H}_k$ has $n$ vertices and $m$ edges, then we prove that $k(k^2 - 3)\tau(H) \le (k-2)(k+1)n + (k - 1)^2m + k-1$ and $k(k^2 - 3)\alpha(H) \ge  (k^2 + k - 4)n  - (k-1)^2 m - (k-1)$. Similar bounds are proven in the case when $k \ge 2$ is even.

scholarly journals On the α-spectral radius of graphs

2020 ◽  
pp. 22-22
Author(s):  
Haiyan Guo ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Spectral Properties ◽  
Maximum Degree ◽  
Domination Number ◽  
Upper Bounds ◽  
Unicyclic Graphs ◽  
The Family ◽  
The Difference ◽  
Graph Parameters ◽  
Irregular Graphs

For 0 ? ? ? 1, Nikiforov proposed to study the spectral properties of the family of matrices A?(G) = ?D(G)+(1 ? ?)A(G) of a graph G, where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix of G. The ?-spectral radius of G is the largest eigenvalue of A?(G). For a graph with two pendant paths at a vertex or at two adjacent vertices, we prove results concerning the behavior of the ?-spectral radius under relocation of a pendant edge in a pendant path. We give upper bounds for the ?-spectral radius for unicyclic graphs G with maximum degree ? ? 2, connected irregular graphs with given maximum degree and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with the second largest ?-spectral radius among trees, and the unique tree with the largest ?-spectral radius among trees with given diameter. We also determine the unique graphs so that the difference between the maximum degree and the ?-spectral radius is maximum among trees, unicyclic graphs and non-bipartite graphs, respectively.

The Origin of the Moon

1962 ◽  
Vol 14 ◽  
pp. 149-155 ◽  
Author(s):  
E. L. Ruskol
Keyword(s):  
Chemical Composition ◽  
Solar System ◽  
The Moon ◽  
The Earth ◽  
The Difference ◽  
Origin Of The Moon

The difference between average densities of the Moon and Earth was interpreted in the preceding report by Professor H. Urey as indicating a difference in their chemical composition. Therefore, Urey assumes the Moon's formation to have taken place far away from the Earth, under conditions differing substantially from the conditions of Earth's formation. In such a case, the Earth should have captured the Moon. As is admitted by Professor Urey himself, such a capture is a very improbable event. In addition, an assumption that the “lunar” dimensions were representative of protoplanetary bodies in the entire solar system encounters great difficulties.

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