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 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 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 On some mathematical properties of the general zeroth-order Randić coindex of graphs

2020 ◽  
Vol 12 (2) ◽  
pp. 75-82
Author(s):  
I. Milovanović ◽  
M. Matejić ◽  
E. Milovanović ◽  
A. Ali
Keyword(s):  
Real Number ◽  
Lower Bounds ◽  
Connected Graph ◽  
Degree Sequence ◽  
Tree Structure ◽  
Vertex Degree ◽  
Arbitrary Real Number ◽  
Zeroth Order ◽  
Mathematical Properties ◽  
Simple Connected Graph

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n, size m with vertex degree sequence ∆ = d1 ≥ d2 ≥ ··· ≥ dn = d > 0, di = d(vi). Denote by G a complement of G. If vertices vi and v j are adjacent in G, we write i ~ j, otherwise we write i j. The general zeroth-order Randic coindex of ' G is defined as 0Ra(G) = ∑i j (d a-1 i + d a-1 j ) = ∑ n i=1 (n-1-di)d a-1 i , where a is an arbitrary real number. Similarly, general zerothorder Randic coindex of ' G is defined as 0Ra(G) = ∑ n i=1 di(n-1-di) a-1 . New lower bounds for 0Ra(G) and 0Ra(G) are obtained. A case when G has a tree structure is also covered.

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 Edge-Decomposition of Graphs into Copies of a Tree with Four Edges

10.37236/2110 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
János Barát ◽  
Dániel Gerbner
Keyword(s):  
Natural Number ◽  
Connected Graph ◽  
Degree Sequence ◽  
Decomposition Theorem ◽  
Bipartite Graphs ◽  
Connected Graphs ◽  
Edge Decomposition

We study edge-decompositions of highly connected graphs into copies of a given tree. In particular we attack the following conjecture by Barát and Thomassen: for each tree $T$, there exists a natural number $k_T$ such that if $G$ is a $k_T$-edge-connected graph, and $|E(T)|$ divides $|E(G)|$, then $E(G)$ has a decomposition into copies of $T$. As one of our main results it is sufficient to prove the conjecture for bipartite graphs. The same result has been independently obtained by Carsten Thomassen (2013).Let $Y$ be the unique tree with degree sequence $(1,1,1,2,3)$. We prove that if $G$ is a $191$-edge-connected graph of size divisible by $4$, then $G$ has a $Y$-decomposition. This is the first instance of such a theorem, in which the tree is different from a path or a star. Recently Carsten Thomassen proved a more general decomposition theorem for bistars, which yields the same result with a worse constant.

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 Connectedness criteria for graphs by means of omega invariant

Filomat ◽  
2020 ◽  
Vol 34 (2) ◽  
pp. 647-652
Author(s):  
Utkum Sanli ◽  
Feriha Celik ◽  
Sadik Delen ◽  
Ismail Cangul
Keyword(s):  
Graph Theory ◽  
Euler Characteristic ◽  
Connected Graph ◽  
Topological Property ◽  
Degree Sequence ◽  
Direct Information ◽  
Combinatorial Properties ◽  
Cyclomatic Number ◽  
The Common ◽  
Disconnected Graph

A realizable degree sequence can be realized in many ways as a graph. There are several tests for determining realizability of a degree sequence. Up to now, not much was known about the common properties of these realizations. Euler characteristic is a well-known characteristic of graphs and their underlying surfaces. It is used to determine several combinatorial properties of a surface and of all graphs embedded onto it. Recently, last two authors defined a number ? which is invariant for all realizations of a given degree sequence. ? is shown to be related to Euler characteristic and cyclomatic number. Several properties of ? are obtained and some applications in extremal graph theory are done by authors. As already shown, the number gives direct information compared with the Euler characteristic on the realizability, number of realizations, being acyclic or cyclic, number of components, chords, loops, pendant edges, faces, bridges etc. In this paper, another important topological property of graphs which is connectedness is studied by means of ?. It is shown that all graphs with ?(G)?-4 are disconnected, and if ?(G)? -2, then the graph could be connected or disconnected. It is also shown that if the realization is a connected graph and ?(G)=-2, then certainly the graph should be acyclic. Similarly, it is shown that if the realization is a connected graph G and ?(G)? 0, then certainly the graph should be cyclic. Also, the fact that when ?(G)?-4, the components of the disconnected graph could not all be cyclic, and that if all the components of a graph G are cyclic, then ?(G) ? 0 are proven.

Radius, Diameter and the Degree Sequence of a Graph

2015 ◽  
Vol 65 (6) ◽  
Author(s):  
Jaya Percival Mazorodze ◽  
Simon Mukwembi
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Degree Sequence ◽  
Upper Bounds ◽  
Free Graph

AbstractWe give asymptotically sharp upper bounds on the radius and diameter of(i) a connected graph,(ii) a connected triangle-free graph,(iii) a connected C4-free graph of given order, minimum degree, and maximum degree.We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.

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