scholarly journals The Prism of the Acyclic Orientation Graph is Hamiltonian

10.37236/1199 ◽  
1995 ◽  
Vol 2 (1) ◽  
Author(s):  
Gara Pruesse ◽  
Frank Ruskey
Keyword(s):  
Simple Graph ◽  
Single Edge ◽  
Mixed Graph ◽  
Acyclic Orientations ◽  
Acyclic Orientation

Every connected simple graph $G$ has an acyclic orientation. Define a graph ${AO}(G)$ whose vertices are the acyclic orientations of $G$ and whose edges join orientations that differ by reversing the direction of a single edge. It was known previously that ${AO}(G)$ is connected but not necessarily Hamiltonian. However, Squire proved that the square ${AO}(G)^2$ is Hamiltonian. We prove the slightly stronger result that the prism ${AO}(G) \times e$ is Hamiltonian. If $G$ is a mixed graph (some edges directed, but not necessarily all), then ${AO}(G)$ can be defined as before. The graph ${AO}(G)$ is again connected but we give examples showing that the prism is not necessarily Hamiltonian.

scholarly journals On Graphs whose Orientations are Determined by their Hermitian Spectra

10.37236/9640 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Yi Wang ◽  
Bo-Jun Yuan
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Simple Graph ◽  
Equivalence Classes ◽  
Cycle Space ◽  
Mixed Graph ◽  
Cycle Basis ◽  
Underlying Graph ◽  
Mixed Graphs ◽  
Special Cycle

A mixed graph $D$ is obtained from a simple graph $G$, the underlying graph of $D$, by orienting some edges of $G$. A simple graph $G$ is said to be ODHS (all orientations of $G$ are determined by their $H$-spectra) if every two $H$-cospectral graphs in $\mathcal{D}(G)$ are switching equivalent to each other, where $\mathcal{D}(G)$ is the set of all mixed graphs with $G$ as their underlying graph. In this paper, we characterize all bicyclic ODHS graphs and construct infinitely many ODHS graphs whose cycle spaces are of dimension $k$. For a  connected graph $G$ whose cycle space is of dimension $k$, we also obtain an achievable upper bound $2^{2k-1} + 2^{k-1}$ for the number of switching equivalence classes in $\mathcal{D}(G)$, which naturally is an upper bound of the number of  cospectral classes in $\mathcal{D}(G)$. To achieve these, we propose a valid method to estimate the number of switching equivalence classes in $\mathcal{D}(G)$ based on the strong cycle basis, a special cycle basis  introduced in this paper.

scholarly journals Graph Orientations and Linear Extensions.

2014 ◽  
Vol DMTCS Proceedings vol. AT,... (Proceedings) ◽  
Author(s):  
Benjamin Iriarte
Keyword(s):  
Partial Order ◽  
Simple Graph ◽  
Linear Extensions ◽  
Acyclic Orientations ◽  
International Audience ◽  
General Graphs ◽  
Odd Cycles

International audience Given an underlying undirected simple graph, we consider the set of all acyclic orientations of its edges. Each of these orientations induces a partial order on the vertices of our graph, and therefore we can count the number of linear extensions of these posets. We want to know which choice of orientation maximizes the number of linear extensions of the corresponding poset, and this problem is solved essentially for comparability graphs and odd cycles, presenting several proofs. We then provide an inequality for general graphs and discuss further techniques.

scholarly journals Power graphs and exchange property for resolving sets

2019 ◽  
Vol 17 (1) ◽  
pp. 1303-1309 ◽  
Author(s):  
Ghulam Abbas ◽  
Usman Ali ◽  
Mobeen Munir ◽  
Syed Ahtsham Ul Haq Bokhary ◽  
Shin Min Kang
Keyword(s):  
Present Article ◽  
Linear Algebra ◽  
Finite Groups ◽  
Robot Navigation ◽  
Simple Graph ◽  
Exchange Property ◽  
Metric Dimension ◽  
Sufficient Condition ◽  
Resolving Set ◽  
Dihedral Groups

Abstract Classical applications of resolving sets and metric dimension can be observed in robot navigation, networking and pharmacy. In the present article, a formula for computing the metric dimension of a simple graph wihtout singleton twins is given. A sufficient condition for the graph to have the exchange property for resolving sets is found. Consequently, every minimal resolving set in the graph forms a basis for a matriod in the context of independence defined by Boutin [Determining sets, resolving set and the exchange property, Graphs Combin., 2009, 25, 789-806]. Also, a new way to define a matroid on finite ground is deduced. It is proved that the matroid is strongly base orderable and hence satisfies the conjecture of White [An unique exchange property for bases, Linear Algebra Appl., 1980, 31, 81-91]. As an application, it is shown that the power graphs of some finite groups can define a matroid. Moreover, we also compute the metric dimension of the power graphs of dihedral groups.

scholarly journals Spectral Gap of the Largest Eigenvalue of the Normalized Graph Laplacian

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Florentin Münch
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Spectral Gap ◽  
Complete Bipartite Graph ◽  
Graph Laplacian ◽  
New Method ◽  
Single Edge ◽  
Largest Eigenvalue ◽  
Complement Graph ◽  
The Largest Eigenvalue

AbstractWe offer a new method for proving that the maxima eigenvalue of the normalized graph Laplacian of a graph with n vertices is at least $$\frac{n+1}{n-1}$$ n + 1 n - 1 provided the graph is not complete and that equality is attained if and only if the complement graph is a single edge or a complete bipartite graph with both parts of size $$\frac{n-1}{2}$$ n - 1 2 . With the same method, we also prove a new lower bound to the largest eigenvalue in terms of the minimum vertex degree, provided this is at most $$\frac{n-1}{2}$$ n - 1 2 .

scholarly journals The Irregularity and Modular Irregularity Strength of Fan Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 605
Author(s):  
Martin Bača ◽  
Zuzana Kimáková ◽  
Marcela Lascsáková ◽  
Andrea Semaničová-Feňovčíková
Keyword(s):  
Simple Graph ◽  
Isolated Vertex ◽  
Positive Integers ◽  
Irregularity Strength

For a simple graph G with no isolated edges and at most, one isolated vertex, a labeling φ:E(G)→{1,2,…,k} of positive integers to the edges of G is called irregular if the weights of the vertices, defined as wtφ(v)=∑u∈N(v)φ(uv), are all different. The irregularity strength of a graph G is known as the maximal integer k, minimized over all irregular labelings, and is set to ∞ if no such labeling exists. In this paper, we determine the exact value of the irregularity strength and the modular irregularity strength of fan graphs.

scholarly journals Modelling the Variability and the Anisotropic Behaviour of Crack Growth in SLM Ti-6Al-4V

Materials ◽  
2021 ◽  
Vol 14 (6) ◽  
pp. 1400
Author(s):  
Rhys Jones ◽  
Calvin Rans ◽  
Athanasios P. Iliopoulos ◽  
John G. Michopoulos ◽  
Nam Phan ◽  
...  
Keyword(s):  
Crack Growth ◽  
The United States ◽  
Fatigue Threshold ◽  
Growth Equation ◽  
Single Edge ◽  
Test Program ◽  
Large Variability ◽  
Single Edge Notch ◽  
Anisotropic Behaviour ◽  
Edge Notch

The United States Air Force (USAF) Guidelines for the Durability and Damage Tolerance (DADT) certification of Additive Manufactured (AM) parts states that the most difficult challenge for the certification of an AM part is to establish an accurate prediction of its DADT. How to address this challenge is the focus of the present paper. To this end this paper examines the variability in crack growth in tests on additively manufactured (AM) Ti-6Al-4V specimens built using selective layer melting (SLM). One series of tests analysed involves thirty single edge notch tension specimens with five build orientations and two different post heat treatments. The other test program analysed involved ASTM standard single edge notch specimens with three different build directions. The results of this study highlight the ability of the Hartman–Schijve crack growth equation to capture the variability and the anisotropic behaviour of crack growth in SLM Ti-6Al-4V. It is thus shown that, despite the large variability in crack growth, the intrinsic crack growth equation remains unchanged and that the variability and the anisotropic nature of crack growth in this test program is captured by allowing for changes in both the fatigue threshold and the cyclic fracture toughness.

Tiered Sampling

2021 ◽  
Vol 15 (5) ◽  
pp. 1-52
Author(s):  
Lorenzo De Stefani ◽  
Erisa Terolli ◽  
Eli Upfal
Keyword(s):  
Large Scale ◽  
Analysis Of Algorithms ◽  
Base Layer ◽  
Single Edge ◽  
Real World Data ◽  
High Quality ◽  
Large Graphs ◽  
Massive Graphs ◽  
Variance Estimate ◽  
Low Probability

We introduce Tiered Sampling , a novel technique for estimating the count of sparse motifs in massive graphs whose edges are observed in a stream. Our technique requires only a single pass on the data and uses a memory of fixed size M , which can be magnitudes smaller than the number of edges. Our methods address the challenging task of counting sparse motifs—sub-graph patterns—that have a low probability of appearing in a sample of M edges in the graph, which is the maximum amount of data available to the algorithms in each step. To obtain an unbiased and low variance estimate of the count, we partition the available memory into tiers (layers) of reservoir samples. While the base layer is a standard reservoir sample of edges, other layers are reservoir samples of sub-structures of the desired motif. By storing more frequent sub-structures of the motif, we increase the probability of detecting an occurrence of the sparse motif we are counting, thus decreasing the variance and error of the estimate. While we focus on the designing and analysis of algorithms for counting 4-cliques, we present a method which allows generalizing Tiered Sampling to obtain high-quality estimates for the number of occurrence of any sub-graph of interest, while reducing the analysis effort due to specific properties of the pattern of interest. We present a complete analytical analysis and extensive experimental evaluation of our proposed method using both synthetic and real-world data. Our results demonstrate the advantage of our method in obtaining high-quality approximations for the number of 4 and 5-cliques for large graphs using a very limited amount of memory, significantly outperforming the single edge sample approach for counting sparse motifs in large scale graphs.

scholarly journals k-Zumkeller labeling of super subdivision of some graphs

2021 ◽  
Vol 29 (1) ◽  
Author(s):  
M. Basher
Keyword(s):  
Simple Graph ◽  
Natural Numbers ◽  
Injective Function ◽  
Planar Grid ◽  
Circular Ladder

AbstractA simple graph $$G=(V,E)$$ G = ( V , E ) is said to be k-Zumkeller graph if there is an injective function f from the vertices of G to the natural numbers N such that when each edge $$xy\in E$$ x y ∈ E is assigned the label f(x)f(y), the resulting edge labels are k distinct Zumkeller numbers. In this paper, we show that the super subdivision of path, cycle, comb, ladder, crown, circular ladder, planar grid and prism are k-Zumkeller graphs.

scholarly journals Total Roman {3}-Domination: The Complexity and Linear-Time Algorithm for Trees

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 293
Author(s):  
Xinyue Liu ◽  
Huiqin Jiang ◽  
Pu Wu ◽  
Zehui Shao
Keyword(s):  
Linear Time ◽  
Minimum Weight ◽  
Domination Number ◽  
Time Algorithm ◽  
Simple Graph ◽  
Linear Time Algorithm ◽  
Domination Problem ◽  
Np Complete ◽  
Chordal Bipartite Graphs ◽  
Dominating Function

For a simple graph G=(V,E) with no isolated vertices, a total Roman {3}-dominating function(TR3DF) on G is a function f:V(G)→{0,1,2,3} having the property that (i) ∑w∈N(v)f(w)≥3 if f(v)=0; (ii) ∑w∈N(v)f(w)≥2 if f(v)=1; and (iii) every vertex v with f(v)≠0 has a neighbor u with f(u)≠0 for every vertex v∈V(G). The weight of a TR3DF f is the sum f(V)=∑v∈V(G)f(v) and the minimum weight of a total Roman {3}-dominating function on G is called the total Roman {3}-domination number denoted by γt{R3}(G). In this paper, we show that the total Roman {3}-domination problem is NP-complete for planar graphs and chordal bipartite graphs. Finally, we present a linear-time algorithm to compute the value of γt{R3} for trees.

Sign in / Sign up

Export Citation Format

Share Document