Set Of Graphs Research Articles

An expressive ansatz for low-depth quantum approximate optimisation

The quantum approximate optimisation algorithm (QAOA) is a hybrid quantum–classical algorithm used to approximately solve combinatorial optimisation problems. It involves multiple iterations of a parameterised ansatz that consists of a problem and mixer Hamiltonian, with the parameters being classically optimised. While QAOA can be implemented on near-term quantum hardware, physical limitations such as gate noise, restricted qubit connectivity, and state-preparation-and-measurement (SPAM) errors can limit circuit depth and decrease performance. To address these limitations, this work introduces the eXpressive QAOA (XQAOA), an overparameterised variant of QAOA that assigns more classical parameters to the ansatz to improve its performance at low depths. XQAOA also introduces an additional Pauli-Y component in the mixer Hamiltonian, allowing the mixer to implement arbitrary unitary transformations on each qubit. To benchmark the performance of XQAOA at unit depth, we derive its closed-form expression for the MaxCut problem and compare it to QAOA, Multi-Angle QAOA (MA-QAOA) (Herrman et al 2022 Sci. Rep. 12 6781), a classical-relaxed algorithm, and the state-of-the-art Goemans–Williamson algorithm on a set of unweighted regular graphs with 128 and 256 nodes for degrees ranging from 3 to 10. Our results indicate that at unit depth, XQAOA has benign loss landscapes with local minima concentrated near the global optimum, allowing it to consistently outperform QAOA, MA-QAOA, and the classical-relaxed algorithm on all graph instances and the Goemans–Williamson algorithm on graph instances with degrees greater than 4. Small-scale simulations also reveal that unit-depth XQAOA invariably surpasses both QAOA and MA-QAOA on all tested depths up to five. Additionally, we find an infinite family of graphs for which XQAOA solves MaxCut exactly and analytically show that for some graphs in this family, special cases of XQAOA are capable of achieving a much larger approximation ratio than QAOA. Overall, XQAOA is a more viable choice for variational quantum optimisation on near-term quantum devices, offering competitive performance at low depths.

The Time Complexity of Consensus Under Oblivious Message Adversaries

We study the problem of solving consensus in synchronous directed dynamic networks, in which communication is controlled by an oblivious message adversary that picks the communication graph to be used in a round from a fixed set of graphs D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{D}$$\\end{document} arbitrarily. In this fundamental model, determining consensus solvability and designing efficient consensus algorithms is surprisingly difficult. Enabled by a decision procedure that is derived from a well-established previous consensus solvability characterization for a given set D\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{D}$$\\end{document}, we study, for the first time, the time complexity of solving consensus in this model: We provide both upper and lower bounds for this time complexity, and also relate it to the number of iterations required by the decision procedure. Among other results, we find that reaching consensus under an oblivious message adversary can take exponentially longer than both deciding consensus solvability and broadcasting the input value of some unknown process to all other processes.

Differentially Private Guarantees for Analytics and Machine Learning on Graphs: A Survey of Results

We study the applications of differential privacy (DP) in the context of graph-structured data and discuss the formulations of DP applicable to the publication of graphs and their associated statistics as well as machine learning on graph-based data, including graph neural networks (GNNs). Interpreting DP guarantees in the context of graph-structured data can be challenging, as individual data points are interconnected (often non-linearly or sparsely). This connectivity complicates the computation of individual privacy loss in differentially private learning. The problem is exacerbated by an absence of a single, well-established formulation of DP in graph settings. This issue extends to the domain of GNNs, rendering private machine learning on graph-structured data a challenging task. A lack of prior systematisation work motivated us to study graph-based learning from a privacy perspective. In this work, we systematise different formulations of DP on graphs, discuss challenges and promising applications, including the GNN domain. We compare and separate works into graph analytics tasks and graph learning tasks with GNNs. We conclude our work with a discussion of open questions and potential directions for further research in this area.

Interaction graph engineering in trapped-ion quantum simulators with global drives

Trapped-ion quantum simulators have demonstrated a long history of studying the physics of interacting spin-lattice systems using globally addressed entangling operations. Yet despite the multitude of studies so far, most have been limited to studying variants of the same spin interaction model, namely an Ising model with power-law decay in the couplings. Here, we demonstrate that much broader classes of effective spin–spin interactions are achievable using exclusively global driving fields. Specifically, we find that these new categories of interaction graphs become achievable with perfect or near-perfect theoretical fidelity by tailoring the coupling of the driving fields to each vibrational mode of the ion crystal. Given the relation between the ion crystal vibrational modes and the accessible interaction graphs, we show how the accessible interaction graph set can be further expanded by shaping the trapping potential to include specific anharmonic terms. Finally, we derive a rigorous test to determine whether a desired interaction graph is accessible using only globally driven fields. These tools broaden the reach of trapped-ion quantum simulators so that they may more easily address open questions in materials science and quantum chemistry.

Vertex Cover Hop Dominating Sets in Graphs

Let $G$ be a graph. Then a subset $C$ of vertices of $G$ is called a vertex cover hop dominating if $C$ is both a vertex cover and a hop dominating of $G$. The vertex cover hop domination number of $G$, denoted by $\gamma_{vch}(G)$, is the minimum cardinality among all vertex cover hop dominating sets in $G$. In this paper, we initiate the study of vertex cover hop domination in a graph and we determine its relations with other parameters in graph theory. We characterize the vertex cover hop dominating sets in some special graphs, join, and corona of two graphs and we finally obtain the exact values or bounds of the parameters of these graphs.

Clustering Uniswap v3 traders from their activity on multiple liquidity pools, via novel graph embeddings

Uniswap is a Constant Product Market Maker built around liquidity pools, where pairs of tokens are exchanged subject to a fee that is proportional to the size of transactions. At the time of writing, there exist more than 6000 pools associated with Uniswap v3, implying that empirical investigations on the full ecosystem can easily become computationally expensive. We propose a systematic workflow to extract a tractable sub-universe of liquidity pools, where the interconnection among such pools is maximised to capture broader dynamics within the ecosystem. The resultant set of 34 pools is then used to cluster market participants according to their liquidity consumption behaviour over such environments, for the time window January–June 2022. Introducing a novel approach, we proceed to represent each liquidity taker by a suitably constructed transaction graph. The graph is a fully connected network where nodes are the liquidity taker’s executed transactions on the 34 pools of reference, and edges contain weights encoding the time elapsed between any two transactions. We then extend the NLP-inspired graph2vec algorithm to the weighted undirected setting, and employ it to obtain an embedding of the set of graphs representing market participants. This embedding allows us to extract seven clusters of liquidity takers, with equivalent behavioural patterns that can be interpreted in terms of trading attributes, i.e. preference for exotic assets over stablecoins, frequency of activity, tolerance for higher trading fees.

Himpunan Kritis pada Graf Bintang

Labeling is a one-to-one mapping that maps each element of a graph to Positive numbers called labels. One of its kind is edge-magic total labeling. Under special conditions, the results set of labeled graphs whose subsets are labeled and positioned, which builds the same graph as the labeling, is called the critical set. To obtain the critical set of a graph we must know the type of graph. In this study is a star graph. This study aims to determine the critical set in star graphs. The star graph used is the K1.5 star graph using center points 1, n + 1 and 2n + 1. The research results show that by labeling the total magic side of the K1.5 star graph with center point λ(c) = 1, the magic number k=14 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = n+ 1, the magic number k=18 is obtained. The possible critical set of K1.5 graphs is 120. In the K1.5 Star Graph with center point λ(c) = 2n + 1, the magic number k=22 is obtained. The possible critical set of K1.5 graphs is 120.

Quantum computing dataset of maximum independent set problem on king lattice of over hundred Rydberg atoms

Finding the maximum independent set (MIS) of a large-size graph is a nondeterministic polynomial-time (NP)-complete problem not efficiently solvable with classical computations. Here, we present a set of quantum adiabatic computing data of Rydberg-atom experiments performed to solve the MIS problem of up to 141 atoms randomly arranged on the king lattice. A total of 582,916 events of Rydberg-atom measurements are collected for experimental MIS solutions of 733,853 different graphs. We provide the raw image data along with the entire binary determinations of the measured many-body ground states and the classified graph data, to offer bench-mark testing and advanced data-driven analyses for validation of the performance and system improvements of the Rydberg-atom approach.

Injective Eccentric Domination in Graphs

The concept of domination has inspired researchers which has contributed to a vast literature on domination. A subset of is said to be a dominating set, if every vertex not in is adjacent to at least one vertex in . The eccentricity of is the distance to a vertex farthest from . Thus . For a vertex each vertex at a distance from is an eccentric vertex. The eccentric set of a vertex is defined as . Let , then is known as an eccentric point set of if for every , has at least one vertex such that . A dominating set is called an eccentric dominating set if it is also an eccentric point set. In this article the concept of injective eccentric domination is introduced for simple, connected and undirected graphs. An eccentric dominating set is called an injective eccentric dominating set if for every vertex there exists a vertex such that where is the set of vertices different from and , that are adjacent to both and . Theorems to determine the exact injective eccentric domination number for the basic class of graphs are stated and proved. Nordhaus-Gaddum results are proposed. The injective eccentric dominating set, injective eccentric domination number , upper injective eccentric dominating set and upper injective eccentric domination number for different standard graphs are tabulated.

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdős-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on {1,2,…,N}, there exists a monochromatic red s-clique or a monochromatic blue increasing path Pn with n vertices, provided N>(s−1)(n−1). Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound N>2c(logn)s−1. For the tth power Pnt of the ordered increasing graph path with n vertices, we prove a near linear bound cn(logn)s−2 which improves the previous bound that applied to a more general class of graphs than Pnt due to Conlon-Fox-Lee-Sudakov.

Weakly connected resolving sets in graphs

Harary and Melter introduced the concept of resolving sets in graphs. Slater stated the term locating set and the minimum resolving set as reference sets. Chartrand mentioned the locating set as a resolving set and the minimum cardinality of a resolving set as metric dimension or simply dimension in later stage. Several types of resolving sets like connected resolving sets, independent resolving sets, acyclic resolving sets, etc. have been discussed in the past. In this work, weakly connected resolving set is introduced and we studied its bounds.

Polymorphism of Bis(benzimidazole)bis(thiocyanato-N)cobalt(II) and Its Relevance to Studies of the Chief Color Test for Cocaine

Cobalt(II) thiocyanate-based tests are routinely used to screen cocaine products, with the formation of a blue species interpreted as a positive response. Two popular candidates for the origin of the blue color are an ionic coordination compound, frequently referred to as an ion pair, of the general form (HL)2[Co(SCN)4] or the coordination compound [CoL2(SCN)2], where L represents select nitrogenous bases. Given the high number of nitrogenous bases documented to yield false positives for cobalt(II) thiocyanate-based tests, a reasonable hypothesis is that both candidates are possible but their preferential formation depends on the specific nitrogenous bases screened. This hypothesis was tested through the crystallographic and spectroscopic analysis of reaction products of cocaine hydrochloride, lidocaine monohydrate hydrochloride, and benzimidazole exposed to a classic cobalt(II) thiocyanate reagent. Single-crystal X-ray diffraction revealed that the blue product isolated from benzimidazole test vessels is a coordination compound, with comparative ultraviolet–visible and Raman spectroscopy validating that blue precipitates collected from cocaine hydrochloride and lidocaine monohydrate hydrochloride reaction containers are ionic coordination compounds. Peaks corresponding to π-π* transitions in UV-vis at around 320 nm (cocaine hydrochloride: 320 nm, lidocaine hydrochloride: 323 nm) shift to a higher wavelength of 332 nm for the coordinated benzimidazole, and the broader d-d transitions at 550–630 nm show both a shift and change in envelope for benzimidazole coordinated with cobalt(II). The compound is a new polymorph of bis(benzimidazole)bis(thiocyanato-N)Cobalt(II), γ-[Co(Hbzim)2(SCN)2] (Hbzim = benzimidazole), and the differences in the intermolecular interactions to the two previous polymorphs were clarified by graph set analysis and Hirshfeld surface analysis. Furthermore, the coordination of aromatic nitrogen bases (such as benzimidazole) with Co(II) and aliphatic bases was compared by analyzing the Cambridge Structural Database, and the aromatic bases were found to have a shorter Co-N bond length compared to the aliphatic bases by around 0.02 Å.

Enumeration of dissociation sets in grid graphs

<abstract><p>A dissociation set of a graph $ G $ refers to a set of vertices inducing a subgraph with maximum degree at most 1 and serves as a generalization of two fundamental concepts in graph theory: Independent sets and induced matchings. The enumeration of specific substructures in grid graphs has been a captivating area of research in graph theory. Over the past few decades, the enumeration problems related to various structures in grid graphs such as Hamiltonian cycles, Hamiltonian paths, independent sets, maximal independent sets, and dominating sets have been deeply studied. In this paper, we enumerated dissociation sets in grid graphs using the state matrix recursion algorithm.</p></abstract>

Lower general position sets in graphs

Lower general position sets in graphs

A Central Local Metric Dimension of Generalized Fan Graph, Generalized Broken Fan Graph, and Cm ⊙ K¯m

The central local metric dimension is a new variation of local metric dimension that introduced in 2023. The central local metric dimension is a new concept that enriches research studies in graph theory, especially in the field of metric dimension. This concept combines the concept of local metric dimension by involving central points in the local metric set so that the existence of central points can strengthen the position of the local metric set in distinguishing every two neighboring points in a graph. The methodology of this research is study literature and observation. We find the central vertex of each graph and also find the local metric set of its graph, then we applied it to the related theorem to find the lower bound of central local metric dimension. Let 𝐺 be a connected graph with order 𝑛 and vertex set is 𝑉(𝐺). A subset 𝑊={𝑥1,𝑥2,…,𝑥𝑘}⊆𝑉(𝐺) is a local metric set of graph 𝐺 if the metric code of every two adjacent vertices 𝑢,𝑣 in 𝐺 are 𝑟(𝑢|𝑊)≠𝑟(𝑣|𝑊), where 𝑟(𝑢|𝑊)=(𝑑(𝑢,𝑥1),𝑑(𝑢,𝑥2),…,𝑑(𝑢,𝑥𝑘)) and 𝑟(𝑣|𝑊)=(𝑑(𝑣,𝑥1),𝑑(𝑣,𝑥2),…,𝑑(𝑣,𝑥𝑘)). A vertex 𝑥∈𝑉(𝐺) is a central vertex in 𝐺 if 𝑥 have the the shorthest distance to the another all vertices in 𝐺. If 𝑊 consist of all central vertices in 𝐺, then 𝑊 is called a central local metric set of 𝐺. The minimal cardinality of 𝑊 is called a central basis local set of 𝐺 and its cardinality is called central local metric dimension of 𝐺 or denoted by 𝑙𝑚𝑑𝑠(𝐺). In this paper we explored the central local metric dimension on generalized fan graph, generalized broken fan graph, and a graph resulting from corona operation Cm ⊙ K¯n. The result show that the central local metric dimension of generalized fan graph is equal with its order because the diameter and radius are equal. Different with it, the central local metric dimension of generalized broken fan is equal with its local metric dimension plus cardinality of central set, and the central local metric dimension of Cm ⊙ K¯n are equal with the central local metric dimension of 𝐶𝑚.

Improved covering results for conjugacy classes of symmetric groups via hypercontractivity

Abstract We study covering numbers of subsets of the symmetric group $S_n$ that exhibit closure under conjugation, known as normal sets. We show that for any $\epsilon>0$ , there exists $n_0$ such that if $n>n_0$ and A is a normal subset of the symmetric group $S_n$ of density $\ge e^{-n^{2/5 - \epsilon }}$ , then $A^2 \supseteq A_n$ . This improves upon a seminal result of Larsen and Shalev (Inventiones Math., 2008), with our $2/5$ in the double exponent replacing their $1/4$ . Our proof strategy combines two types of techniques. The first is ‘traditional’ techniques rooted in character bounds and asymptotics for the Witten zeta function, drawing from the foundational works of Liebeck–Shalev, Larsen–Shalev, and more recently, Larsen–Tiep. The second is a sharp hypercontractivity theorem in the symmetric group, which was recently obtained by Keevash and Lifshitz. This synthesis of algebraic and analytic methodologies not only allows us to attain our improved bounds but also provides new insights into the behavior of general independent sets in normal Cayley graphs over symmetric groups.

Construction of an Alexandroff topology on the edge set of an undirected graph

Construction of an Alexandroff topology on the edge set of an undirected graph

Algorithmic Aspects of Vertex-edge Domination in Some Graphs

Let \(G=(V,E)\) be a simple graph. A vertex \(v\in V(G)\) ve-dominates every edge \(uv\) incident to \(v\), as well as every edge adjacent to these incident edges. A set \(D\subseteq V(G)\) is a vertex-edge dominating set if every edge of \(E(G)\) is ve-dominated by a vertex of \(D.\) The MINIMUM VERTEX-EDGE DOMINATION problem is to find a vertex-edge dominating set of minimum cardinality. A linear time algorithm to find the minimum vertex-edge dominating set for proper interval graphs is proposed. The vertex-edge domination problem is proved to be APX-complete for bounded-free graphs and NP-Complete for Chordal bipartite and Undirected Path graphs.

IPCC7: Post-Quantum Encryption Scheme Based on a Perfect Dominating Set in 3-Regular Graph

IPCC7: Post-Quantum Encryption Scheme Based on a Perfect Dominating Set in 3-Regular Graph

Cardinality bounds on subsets in the partition resolving set for complex convex polytope-like graph

<abstract><p>Let $ G = (V, E) $ be a simple, connected graph with vertex set $ V(G) $ and $ E(G) $ edge set of $ G $. For two vertices $ a $ and $ b $ in a graph $ G $, the distance $ d(a, b) $ from $ a $ to $ b $ is the length of shortest path $ a-b $ path in $ G $. A $ k $-ordered partition of vertices of $ G $ is represented as $ {R}{p} = \{{R}{p_1}, {R}{p_2}, \dots, {R}{p_k}\} $ and the representation $ r(a|{R}{p}) $ of a vertex $ a $ with respect to $ {R}{p} $ is the vector $ (d(a|{R}{p_1}), d(a|{R}{p_2}), \dots, d(a|{R}{p_k})) $. The partition is called a resolving partition of $ G $ if $ r(a|{R}{p}) \ne r(b|{R}{p}) $ for all distinct $ a, b\in V(G) $. The partition dimension of a graph, denoted by $ pd(G) $, is the cardinality of a minimum resolving partition of $ G $. Computing precise and constant values for the partition dimension poses a interesting problem; therefore, it is possible to compute an upper bound for the partition dimension within a general family of graphs. In this paper, we studied partition dimension of the some families of convex polytopes, specifically $ \mathbb{T}_n $, $ \mathbb{U}_n $, $ \mathbb{V}_n $, and $ \mathbb{A}_n $, and proved that these graphs have constant partition dimension.</p></abstract>