scholarly journals Every line graph of a 4-edge-connected graph is Z3-connected

2009 ◽  
Vol 30 (2) ◽  
pp. 595-601 ◽  
Author(s):  
Hong-Jian Lai ◽  
Lianying Miao ◽  
Yehong Shao
Keyword(s):  
Connected Graph ◽  
Line Graph

scholarly journals Unanimous and Strategy-Proof Probabilistic Rules for Single-Peaked Preference Profiles on Graphs

2021 ◽  
Author(s):  
Hans Peters ◽  
Souvik Roy ◽  
Soumyarup Sadhukhan
Keyword(s):  
Probability Distribution ◽  
Connected Graph ◽  
Line Graph ◽  
Preference Profile ◽  
Tree Representation ◽  
General Characterization ◽  
Finite Set ◽  
Probabilistic Rule ◽  
Strategy Proof ◽  
Block Tree

Finitely many agents have preferences on a finite set of alternatives, single-peaked with respect to a connected graph with these alternatives as vertices. A probabilistic rule assigns to each preference profile a probability distribution over the alternatives. First, all unanimous and strategy-proof probabilistic rules are characterized when the graph is a tree. These rules are uniquely determined by their outcomes at those preference profiles at which all peaks are on leaves of the tree and, thus, extend the known case of a line graph. Second, it is shown that every unanimous and strategy-proof probabilistic rule is random dictatorial if and only if the graph has no leaves. Finally, the two results are combined to obtain a general characterization for every connected graph by using its block tree representation.

scholarly journals Some Results on the Eccentric Sequence of Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 55-57
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Degree Sequence ◽  
Integral Graph

The distance d(u, v) from a vertex u of graph G to a vertex v is the length of a shortest u to v path. The degree of the vertex u is the number of vertices at distance one. The sequence of numbers of vertices having 0,1,2,3,... is called the degree sequence, which is the list of degrees of vertices of G arranged in non-decreasing order. The eccentricity e(a) of a is the distance of a farthest vertex from a. Let G be a connected graph. The Eccentric Sequence of G is the list of the eccentricities of its vertices arranged in non-decreasing order. In this paper, we characterize the eccentric sequence of some of the derived graphs namely the line graph of the integral graph , the eccentric sequence of Mycieleskian of a graph.

scholarly journals Computing Minimal Doubly Resolving Sets and the Strong Metric Dimension of the Layer Sun Graph and the Line Graph of the Layer Sun Graph

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Jia-Bao Liu ◽  
Ali Zafari
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality ◽  
Strong Metric Dimension ◽  
Vertex Set

Let G be a finite, connected graph of order of, at least, 2 with vertex set VG and edge set EG. A set S of vertices of the graph G is a doubly resolving set for G if every two distinct vertices of G are doubly resolved by some two vertices of S. The minimal doubly resolving set of vertices of graph G is a doubly resolving set with minimum cardinality and is denoted by ψG. In this paper, first, we construct a class of graphs of order 2n+Σr=1k−2nmr, denoted by LSGn,m,k, and call these graphs as the layer Sun graphs with parameters n, m, and k. Moreover, we compute minimal doubly resolving sets and the strong metric dimension of the layer Sun graph LSGn,m,k and the line graph of the layer Sun graph LSGn,m,k.

scholarly journals Maximum Degree Growth of the Iterated Line Graph

10.37236/1460 ◽  
1999 ◽  
Vol 6 (1) ◽  
Author(s):  
Stephen G. Hartke ◽  
Aparna W. Higgins
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Line Graph ◽  
Induced Subgraphs ◽  
Iterated Line Graph

Let $\Delta_k$ denote the maximum degree of the $k^{\rm th}$ iterated line graph $L^k(G)$. For any connected graph $G$ that is not a path, the inequality $\Delta_{k+1}\leq 2\Delta_k-2$ holds. Niepel, Knor, and Šoltés have conjectured that there exists an integer $K$ such that, for all $k\geq K$, equality holds; that is, the maximum degree $\Delta_k$ attains the greatest possible growth. We prove this conjecture using induced subgraphs of maximum degree vertices and locally maximum vertices.

scholarly journals Domination Integrity of Some Special Graphs

2020 ◽  
Vol 8 (4S5) ◽  
pp. 40-45
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Dominating Set ◽  
The Stability

A major importance is given to the stability of a network by its users and designers. Domination Integrity is one of the parameter used to determine the network’s vulnerability and its defined for a connected graph G as “ DI(G) = min{| X | +m(G - X) : X is a dominating set} wh ere m(G-X) is the order of maximum component of G-X”, by Sundareswaran and Swaminathan. Here we investigate the domination integrity of Tadpole graph, Lollipop graph and a line graph of composition of nP and P2

scholarly journals On The Local Metric Dimension of Line Graph of Special Graph

CAUCHY ◽  
2016 ◽  
Vol 4 (3) ◽  
pp. 125
Author(s):  
Marsidi Marsidi ◽  
Dafik Dafik ◽  
Ika Hesti Agustin ◽  
Ridho Alfarisi
Keyword(s):  
Connected Graph ◽  
Ordered Set ◽  
Line Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Minimum Cardinality

Let G be a simple, nontrivial, and connected graph.  is a representation of an ordered set of <em>k</em> distinct vertices in a nontrivial connected graph G. The metric code of a vertex <em>v</em>, where <em>, </em>the ordered  of <em>k</em>-vector is representations of <em>v</em> with respect to <em>W</em>, where  is the distance between the vertices <em>v</em> and <em>w<sub>i</sub></em> for 1≤ <em>i ≤k</em>.  Furthermore, the set W is called a local resolving set of G if  for every pair <em>u</em>,<em>v </em>of adjacent vertices of G. The local metric dimension ldim(G) is minimum cardinality of <em>W</em>. The local metric dimension exists for every nontrivial connected graph G. In this paper, we study the local metric dimension of line graph of special graphs , namely path, cycle, generalized star, and wheel. The line graph L(G) of a graph G has a vertex for each edge of G, and two vertices in L(G) are adjacent if and only if the corresponding edges in G have a vertex in common.

On the line Graph of a Finite Affine Plane

1965 ◽  
Vol 17 ◽  
pp. 687-694 ◽  
Author(s):  
A. J. Hoffman ◽  
D. K. Ray-Chaudhuri
Keyword(s):  
Connected Graph ◽  
Affine Plane ◽  
Line Graph ◽  
The Other ◽  
Finite Affine

Let n be a finite affine plane with n points on a line. We denote by G(II) the graph whose vertices are all points and lines of II, with two vertices adjacent if and only if one is a point, the other is a line, and the point and line are incident. Let L(11) denote the line graph of G(II), i.e., the vertices of L(II) are the edges of G(II), and two vertices of L(II) are adjacent if the corresponding edges of G(II) are adjacent. It is clear that L(II) is a regular, connected graph with n2(n + 1) vertices and valence 2n — 1.

scholarly journals On the Edge Wiener index

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 541-549
Author(s):  
Abolghasem Soltani ◽  
Ali Iranmanesh
Keyword(s):  
Explicit Formula ◽  
Connected Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Wiener Index ◽  
New Method ◽  
Automorphisms Of Graphs ◽  
Group Automorphisms ◽  
Sum Of Distances ◽  
Simple Connected Graph

Let G be a simple connected graph. The Wiener index of G is the sum of all distances between vertices of G. Whereas, the edge Wiener index of G is defined as the sum of distances between all pairs of edges of G where the distance between the edges f and g in E(G) is defined as the distance between the vertices f and g in the line graph of G. In this paper we will describe a new method for calculating the edge Wiener index. Then find this index for the triangular graphs. Also, we obtain an explicit formula for the Wiener index of the Cartesian product of two graphs using the group automorphisms of graphs.

scholarly journals On the edge-Wiener index of the disjunctive product of simple graphs

2020 ◽  
Vol 30 (1) ◽  
pp. 1-14
Author(s):  
M. Azari ◽  
◽  
A. Iranmanesh ◽  
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Wiener Index ◽  
Simple Graphs ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs

The edge-Wiener index of a simple connected graph G is defined as the sum of distances between all pairs of edges of G where the distance between two edges in G is the distance between the corresponding vertices in the line graph of G. In this paper, we study the edge-Wiener index under the disjunctive product of graphs and apply our results to compute the edge-Wiener index for the disjunctive product of paths and cycles.

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