scholarly journals On a problem of ore on maximal trees

1970 ◽  
Vol 11 (3) ◽  
pp. 379-380 ◽  
Author(s):  
S. B. Rao
Keyword(s):  
Connected Graph ◽  
Maximal Tree ◽  
Finite Case ◽  
Multiple Edges

We consider only graphs without loops or multiple edges. Pertinent definitions are given below. For notation and other definitions we generally follow Ore [1].A connected graph G = (X, E) is said to have the property P if for every maximal tree T of G there exists a vertex aT of G such that distance between aT and x is same in T as in G for every x in X. The following problem has been posed by Ore (see [1] page 103, problem 4): Determine the graphs with property P. This paper presents a solution to the above problem in the finite case.

On the Minimal Graph with a Given Number of Spanning Trees

1970 ◽  
Vol 13 (4) ◽  
pp. 515-517 ◽  
Author(s):  
J. Sedláček
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Private Communication ◽  
Minimal Graph ◽  
Maximal Tree ◽  
Multiple Edges ◽  
Number Of Spanning Trees

Let G be a finite connected graph without loops or multiple edges. A maximal tree subgraph T of G is called a spanning tree of G. Denote by k(G) the number of all trees spanning the graph G. A. Rosa formulated the following problem (private communication): Let x(≠2) be a given positive integer and denote by α(x) the smallest positive integer y having the following property: There exists a graph G on y vertices with x spanning trees. Investigate the behavior of the function α(x).

scholarly journals Undirecting membership in models of Anti-Foundation

2020 ◽  
Author(s):  
Bea Adam-Day ◽  
Peter J. Cameron
Keyword(s):  
Set Theory ◽  
Random Graph ◽  
Connected Graph ◽  
Countable Model ◽  
Connected Components ◽  
Membership Relation ◽  
Multiple Edges

AbstractIt is known that, if we take a countable model of Zermelo–Fraenkel set theory ZFC and “undirect” the membership relation (that is, make a graph by joining x to y if either $$x\in y$$ x ∈ y or $$y\in x$$ y ∈ x ), we obtain the Erdős–Rényi random graph. The crucial axiom in the proof of this is the Axiom of Foundation; so it is natural to wonder what happens if we delete this axiom, or replace it by an alternative (such as Aczel’s Anti-Foundation Axiom). The resulting graph may fail to be simple; it may have loops (if $$x\in x$$ x ∈ x for some x) or multiple edges (if $$x\in y$$ x ∈ y and $$y\in x$$ y ∈ x for some distinct x, y). We show that, in ZFA, if we keep the loops and ignore the multiple edges, we obtain the “random loopy graph” (which is $$\aleph _0$$ ℵ 0 -categorical and homogeneous), but if we keep multiple edges, the resulting graph is not $$\aleph _0$$ ℵ 0 -categorical, but has infinitely many 1-types. Moreover, if we keep only loops and double edges and discard single edges, the resulting graph contains countably many connected components isomorphic to any given finite connected graph with loops.

scholarly journals Π(G,x) Polynomial and Π(G) Index of Armchair Polyhex Nanotubes TUAC6[m,n]

2014 ◽  
Vol 36 ◽  
pp. 201-206
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Infinite Class ◽  
Vertex Set ◽  
Omega Polynomial ◽  
Multiple Edges ◽  
Simple Connected Graph

Let G be a simple connected graph with the vertex set V = V(G) and the edge set E = E(G), without loops and multiple edges. For counting qoc strips in G, Omega polynomial was introduced by Diudea and was defined as Ω(G,x) = ∑cm(G,c)xc where m(G,c) be number of qoc strips of length c in the graph G. Following Omega polynomial, the Sadhana polynomial was defined by Ashrafi et al as Sd(G,x) = ∑cm(G,c)x[E(G)]-c in this paper we compute the Pi polynomial Π(G,x) = ∑cm(G,c)x[E(G)]-c and Pi Index Π(G ) = ∑cc·m(G,c)([E(G)]-c) of an infinite class of “Armchair polyhex nanotubes TUAC6[m,n]”.

Linear-Time Approximation Algorithms for the Max Cut Problem

1993 ◽  
Vol 2 (2) ◽  
pp. 201-210 ◽  
Author(s):  
Nguyen van Ngoc ◽  
Zsolt Tuza
Keyword(s):  
Lower Bound ◽  
Approximation Algorithms ◽  
Lower Bounds ◽  
Connected Graph ◽  
Linear Time ◽  
Worst Case ◽  
Time Approximation ◽  
Bipartite Subgraph ◽  
Max Cut Problem ◽  
Multiple Edges

Let G be a connected graph with n vertices and m edges (multiple edges allowed), and let k ≥ 2 be an integer. There is an algorithm with (optimal) running time of O(m) that finds(i) a bipartite subgraph of G with ≥ m/2 + (n − 1)/4 edges,(ii) a bipartite subgraph of G with ≥ m/2 + 3(n−1)/8 edges if G is triangle-free,(iii) a k-colourable subgraph of G with ≥ m − m/k + (n−1)/k + (k − 3)/2 edges if k ≥ 3 and G is not k-colorable.Infinite families of graphs show that each of those lower bounds on the worst-case performance are best possible (for every algorithm). Moreover, even if short cycles are excluded, the general lower bound of m − m/k cannot be replaced by m − m/k + εm for any fixed ε > 0; and it is NP-complete to decide whether a graph with m edges contains a k-colorable subgraph with more than m − m/k + εm edges, for any k ≥ 2 and ε> 0, ε < 1/k.

scholarly journals Graph Self-Transformation Model Based on the Operation of Change the End of the Edge

2020 ◽  
Vol 23 (3) ◽  
pp. 315-335
Author(s):  
Igor Borisovich Burdonov
Keyword(s):  
Connected Graph ◽  
Undirected Graph ◽  
Wiener Index ◽  
Transformation Model ◽  
Common Vertex ◽  
Distributed Network ◽  
Undirected Graphs ◽  
Limited Degree ◽  
Multiple Edges ◽  
Root Tree

We consider a distributed network whose topology is described by an undirected graph. The network itself can change its topology, using special “commands” provided by its nodes. The work proposes an extremely local atomic transformation acb of a change the end c of the edge ac, “moving” along the edge cb from vertex c to vertex b. As a result of this operation, the edge ac is removed, and the edge ab is added. Such a transformation is performed by a “command” from a common vertex c of two adjacent edges ac and cb. It is shown that from any tree you can get any other tree with the same set of vertices using only atomic transformations. If the degrees of the tree vertices are bounded by the number d (d 3), then the transformation does not violate this restriction. As an example of the purpose of such a transformation, the problems of maximizing and minimizing the Wiener index of a tree with a limited degree of vertices without changing the set of its vertices are considered. The Wiener index is the sum of pairwise distances between the vertices of a graph. The maximum Wiener index has a linear tree (a tree with two leaf vertices). For a root tree with a minimum Wiener index, its type and method for calculating the number of vertices in the branches of the neighbors of the root are determined. Two distributed algorithms are proposed: transforming a tree into a linear tree and transforming a linear tree into a tree with a minimum Wiener index. It is proved that both algorithms have complexity no higher than 2n–2, where n is the number of tree vertices. We also consider the transformation of arbitrary undirected graphs, in which there can be cycles, multiple edges and loops, without restricting the degree of the vertices. It is shown that any connected graph with n vertices can be transformed into any other connected graph with k vertices and the same number of edges in no more than 2(n+k)–2.

scholarly journals On the SD-Polynomial and SD-Index of an Infinite Class of “Armchair Polyhex Nanotubes”

2014 ◽  
Vol 31 ◽  
pp. 63-68
Author(s):  
Mohammad Reza Farahani
Keyword(s):  
Connected Graph ◽  
Infinite Class ◽  
Opposite Edge ◽  
First Derivative ◽  
Vertex Set ◽  
Multiple Edges ◽  
Simple Connected Graph

Let G be a simple connected graph with the vertex set V = V(G) and the edge set E = E(G),without loops and multiple edges. For counting qoc strips in G, Diudea introduced the Ω-polynomialof G and was defined as Ω(G, x) = ∑ki-1xi where C1, C2,..., Ck be the “opposite edge strips” ops of Gand ci = |Ci| (I = 1, 2,..., k). One can obtain the Sd-polynomial by replacing xc with x|E(G)|-c in Ω-polynomial. Then the Sd-index will be the first derivative of Sd(x) evaluated at x = 1. In this paper wecompute the Sd-polynomial and Sd-index of an infinite class of “Armchair Polyhe x Nanotubes”.

scholarly journals A note on PM-compact $ K_4 $-free bricks

2021 ◽  
Vol 7 (3) ◽  
pp. 3648-3652
Author(s):  
Jinqiu Zhou ◽  
◽  
Qunfang Li ◽  
Keyword(s):  
Connected Graph ◽  
Perfect Matching ◽  
Building Blocks ◽  
Discrete Mathematics ◽  
Siam Journal ◽  
Multiple Edges ◽  
Generation Procedure

<abstract><p>A 3-connected graph is a <italic>brick</italic> if the graph obtained from it by deleting any two distinct vertices has a perfect matching. The importance of bricks stems from the fact that they are building blocks of the matching covered graphs. Lovász (Combinatorica, 3 (1983), 105-117) showed that every brick is $ K_4 $-based or $ \overline{C}_6 $-based. A brick is <italic>$ K_4 $-free</italic> (respectively, <italic>$ \overline{C}_6 $-free</italic>) if it is not $ K_4 $-based (respectively, $ \overline{C}_6 $-based). Recently, Carvalho, Lucchesi and Murty (SIAM Journal on Discrete Mathematics, 34(3) (2020), 1769-1790) characterised the PM-compact $ \overline{C}_6 $-free bricks. In this note, we show that, by using the brick generation procedure established by Norine and Thomas (J Combin Theory Ser B, 97 (2007), 769-817), the only PM-compact $ K_4 $-free brick is $ \overline{C}_6 $, up to multiple edges.</p></abstract>

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

Hamiltonicity of a class of toroidal graphs

2020 ◽  
Vol 70 (2) ◽  
pp. 497-503
Author(s):  
Dipendu Maity ◽  
Ashish Kumar Upadhyay
Keyword(s):  
Connected Graph ◽  
Hamiltonian Cycle ◽  
Partial Solution ◽  
The Other ◽  
Hamiltonian Cycles ◽  
The Face ◽  
Toroidal Graphs

Abstract If the face-cycles at all the vertices in a map are of same type then the map is said to be a semi-equivelar map. There are eleven types of semi-equivelar maps on the torus. In 1972 Altshuler has presented a study of Hamiltonian cycles in semi-equivelar maps of three types {36}, {44} and {63} on the torus. In this article we study Hamiltonicity of semi-equivelar maps of the other eight types {33, 42}, {32, 41, 31, 41}, {31, 61, 31, 61}, {34, 61}, {41, 82}, {31, 122}, {41, 61, 121} and {31, 41, 61, 41} on the torus. This gives a partial solution to the well known Conjecture that every 4-connected graph on the torus has a Hamiltonian cycle.

scholarly journals Privacy Preserving Distributed Summation in a Connected Graph

2020 ◽  
Vol 53 (2) ◽  
pp. 3445-3450
Author(s):  
Katrine Tjell ◽  
Rafael Wisniewski
Keyword(s):  
Connected Graph ◽  
Privacy Preserving
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