scholarly journals Antipodal graphs and digraphs

1993 ◽  
Vol 16 (3) ◽  
pp. 579-586 ◽  
Author(s):  
Garry Johns ◽  
Karen Sleno
Keyword(s):  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Vertex Set

The antipodal graph of a graphG, denoted byA(G), has the same vertex set asGwith an edge joining verticesuandvifd(u,v)is equal to the diameter ofG. (IfGis disconnected, thendiam G=∞.) This definition is extended to a digraphDwhere the arc(u,v)is included inA(D)ifd(u,v)is the diameter ofD. It is shown that a digraphDis an antipodal digraph if and only ifDis the antipodal digraph of its complement. This generalizes a known characterization for antipodal graphs and provides an improved proof. Examples and properties of antipodal digraphs are given. A digraphDis self-antipodal ifA(D)is isomorphic toD. Several characteristics of a self-antipodal digraphDare given including sharp upper and lower bounds on the size ofD. Similar results are given for self-antipodal graphs.

scholarly journals Wiener–Hosoya Matrix of Connected Graphs

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 359
Author(s):  
Hassan Ibrahim ◽  
Reza Sharafdini ◽  
Tamás Réti ◽  
Abolape Akwu
Keyword(s):  
Lower Bounds ◽  
Molecular Graph ◽  
Wiener Index ◽  
Vertex Degree ◽  
Upper And Lower Bounds ◽  
Connected Graphs ◽  
Matrix Description ◽  
Vertex Set

Let G be a connected (molecular) graph with the vertex set V(G)={v1,⋯,vn}, and let di and σi denote, respectively, the vertex degree and the transmission of vi, for 1≤i≤n. In this paper, we aim to provide a new matrix description of the celebrated Wiener index. In fact, we introduce the Wiener–Hosoya matrix of G, which is defined as the n×n matrix whose (i,j)-entry is equal to σi2di+σj2dj if vi and vj are adjacent and 0 otherwise. Some properties, including upper and lower bounds for the eigenvalues of the Wiener–Hosoya matrix are obtained and the extremal cases are described. Further, we introduce the energy of this matrix.

scholarly journals ON STRUCTURAL AND GRAPH THEORETIC PROPERTIES OF HIGHER ORDER DELAUNAY GRAPHS

2009 ◽  
Vol 19 (06) ◽  
pp. 595-615 ◽  
Author(s):  
MANUEL ABELLANAS ◽  
PROSENJIT BOSE ◽  
JESÚS GARCÍA ◽  
FERRAN HURTADO ◽  
CARLOS M. NICOLÁS ◽  
...  
Keyword(s):  
Cellular Networks ◽  
Lower Bounds ◽  
Higher Order ◽  
Combinatorial Structure ◽  
Upper And Lower Bounds ◽  
Coloring Problem ◽  
Graph Theoretic ◽  
Convex Position ◽  
Gabriel Graph ◽  
Vertex Set

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k ≥ 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks.

Vertex-Neighbor-Scattering Number of Bipartite Graphs

2016 ◽  
Vol 27 (04) ◽  
pp. 501-509
Author(s):  
Zongtian Wei ◽  
Nannan Qi ◽  
Xiaokui Yue
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Bipartite Graphs ◽  
Upper And Lower Bounds ◽  
Number Of Components ◽  
Np Completeness ◽  
Vertex Set ◽  
Np Complete

Let G be a connected graph. A set of vertices [Formula: see text] is called subverted from G if each of the vertices in S and the neighbor of S in G are deleted from G. By G/S we denote the survival subgraph that remains after S is subverted from G. A vertex set S is called a cut-strategy of G if G/S is disconnected, a clique, or ø. The vertex-neighbor-scattering number of G is defined by [Formula: see text], where S is any cut-strategy of G, and ø(G/S) is the number of components of G/S. It is known that this parameter can be used to measure the vulnerability of spy networks and the computing problem of the parameter is NP-complete. In this paper, we discuss the vertex-neighbor-scattering number of bipartite graphs. The NP-completeness of the computing problem of this parameter is proven, and some upper and lower bounds of the parameter are also given.

scholarly journals Independent Transversals and Independent Coverings in Sparse Partite Graphs

1997 ◽  
Vol 6 (1) ◽  
pp. 115-125 ◽  
Author(s):  
RAPHAEL YUSTER
Keyword(s):  
Lower Bounds ◽  
Pairwise Disjoint ◽  
Independent Set ◽  
Independent Sets ◽  
Upper And Lower Bounds ◽  
Vertex Set

An [n, k, r]-partite graph is a graph whose vertex set, V, can be partitioned into n pairwise-disjoint independent sets, V1, …, Vn, each containing exactly k vertices, and the subgraph induced by Vi ∪ Vj contains exactly r independent edges, for 1 [les ] i < j [les ] n. An independent transversal in an [n, k, r]-partite graph is an independent set, T, consisting of n vertices, one from each Vi. An independent covering is a set of k pairwise-disjoint independent transversals. Let t(k, r) denote the maximal n for which every [n, k, r]-partite graph contains an independent transversal. Let c(k, r) be the maximal n for which every [n, k, r]-partite graph contains an independent covering. We give upper and lower bounds for these parameters. Furthermore, our bounds are constructive. These results improve and generalize previous results of Erdo″s, Gyárfás and Łuczak [5], for the case of graphs.

scholarly journals Two coloring problems on matrix graphs

2016 ◽  
Vol 08 (03) ◽  
pp. 1650053
Author(s):  
Zhe Han ◽  
Mei Lu
Keyword(s):  
Finite Field ◽  
Optical Networks ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Basic Properties ◽  
New Family ◽  
Minimum Number ◽  
Vertex Set ◽  
Positive Integers

In this paper, we propose a new family of graphs, matrix graphs, whose vertex set [Formula: see text] is the set of all [Formula: see text] matrices over a finite field [Formula: see text] for any positive integers [Formula: see text] and [Formula: see text]. And any two matrices share an edge if the rank of their difference is [Formula: see text]. Next, we give some basic properties of such graphs and also consider two coloring problems on them. Let [Formula: see text] (resp., [Formula: see text]) denote the minimum number of colors necessary to color the above matrix graph so that no two vertices that are at a distance at most [Formula: see text] (resp., exactly [Formula: see text]) get the same color. These two problems were proposed in the study of scalability of optical networks. In this paper, we determine the exact value of [Formula: see text] and give some upper and lower bounds on [Formula: see text].

scholarly journals A New Lower Bound on the Density of Vertex Identifying Codes for the Infinite Hexagonal Grid

10.37236/202 ◽  
2009 ◽  
Vol 16 (1) ◽  
Author(s):  
Daniel W. Cranston ◽  
Gexin Yu
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Hexagonal Grid ◽  
Identifying Codes ◽  
Identifying Code ◽  
Vertex Set

Given a graph $G$, an identifying code ${\cal D}\subseteq V(G)$ is a vertex set such that for any two distinct vertices $v_1,v_2\in V(G)$, the sets $N[v_1]\cap{\cal D}$ and $N[v_2]\cap{\cal D}$ are distinct and nonempty (here $N[v]$ denotes a vertex $v$ and its neighbors). We study the case when $G$ is the infinite hexagonal grid $H$. Cohen et.al. constructed two identifying codes for $H$ with density $3/7$ and proved that any identifying code for $H$ must have density at least $16/39\approx0.410256$. Both their upper and lower bounds were best known until now. Here we prove a lower bound of $12/29\approx0.413793$.

scholarly journals Growth Rates of Groups associated with Face 2-Coloured Triangulations and Directed Eulerian Digraphs on the Sphere

10.37236/5410 ◽  
2016 ◽  
Vol 23 (1) ◽  
Author(s):  
Thomas A. McCourt
Keyword(s):  
Abelian Group ◽  
Lower Bounds ◽  
Growth Rates ◽  
Piecewise Linear ◽  
Upper And Lower Bounds ◽  
Torsion Subgroup ◽  
Pile Groups ◽  
Sand Pile ◽  
Black And White ◽  
Vertex Set

Let $\mathcal{G}$ be a properly face $2$-coloured (say black and white) piecewise-linear triangulation of the sphere with vertex set $V$. Consider the abelian group $\mathcal{A}_W$ generated by the set $V$, with relations $r+c+s=0$ for all white triangles with vertices $r$, $c$ and $s$. The group $\mathcal{A}_B$ can be defined similarly, using black triangles. These groups are related in the following manner $\mathcal{A}_W\cong\mathcal{A}_B\cong\mathbb{Z}\oplus\mathbb{Z}\oplus\mathcal{C}$ where $\mathcal{C}$ is a finite abelian group.The finite torsion subgroup $\mathcal{C}$ is referred to as the canonical group of the triangulation. Let $m_t$ be the maximal order of $\mathcal{C}$ over all properly face 2-coloured spherical triangulations with $t$ triangles of each colour. By relating such a triangulation to certain directed Eulerian spherical embeddings of digraphs whose abelian sand-pile groups are isomorphic to the triangulation's canonical group we provide improved upper and lower bounds for $\lim \sup_{t\rightarrow\infty}(m_t)^{1/t}$.

scholarly journals Universal Guard Problems

2018 ◽  
Vol 28 (02) ◽  
pp. 129-160
Author(s):  
Sándor P. Fekete ◽  
Qian Li ◽  
Joseph S. B. Mitchell ◽  
Christian Scheffer
Keyword(s):  
Lower Bounds ◽  
Upper Bound ◽  
Simple Polygon ◽  
Upper And Lower Bounds ◽  
Point Vertex ◽  
Vertex Set

Given a set [Formula: see text] of [Formula: see text] points in the plane, how many universal guards are sometimes necessary and always sufficient to guard any simple polygon with vertex set [Formula: see text]? We call this problem a Universal Guard Problem and provide a spectrum of results. We give upper and lower bounds on the number of universal guards that are always sufficient to guard all polygons having a given set of [Formula: see text] vertices, or to guard all polygons in a given set of [Formula: see text] polygons on an [Formula: see text]-point vertex set. Our upper bound proofs include algorithms to construct universal guard sets of the respective cardinalities.

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.

scholarly journals Quadruple Roman Domination in Trees

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1318
Author(s):  
Zheng Kou ◽  
Saeed Kosari ◽  
Guoliang Hao ◽  
Jafar Amjadi ◽  
Nesa Khalili
Keyword(s):  
Open Neighborhood ◽  
Minimum Weight ◽  
Domination Number ◽  
Theoretical Computer Science ◽  
Upper And Lower Bounds ◽  
Discrete Mathematics ◽  
Roman Domination ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function

This paper is devoted to the study of the quadruple Roman domination in trees, and it is a contribution to the Special Issue “Theoretical computer science and discrete mathematics” of Symmetry. For any positive integer k, a [k]-Roman dominating function ([k]-RDF) of a simple graph G is a function from the vertex set V of G to the set {0,1,2,…,k+1} if for any vertex u∈V with f(u)<k, ∑x∈N(u)∪{u}f(x)≥|{x∈N(u):f(x)≥1}|+k, where N(u) is the open neighborhood of u. The weight of a [k]-RDF is the value Σv∈Vf(v). The minimum weight of a [k]-RDF is called the [k]-Roman domination number γ[kR](G) of G. In this paper, we establish sharp upper and lower bounds on γ[4R](T) for nontrivial trees T and characterize extremal trees.

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