Pair Of Graphs Research Articles

The Multi-Maximum and Quasi-Maximum Common Subgraph Problem

The Maximum Common Subgraph problem has been long proven NP-hard. Nevertheless, it has countless practical applications, and researchers are still searching for exact solutions and scalable heuristic approaches. Driven by applications in molecular science and cyber-security, we concentrate on the Maximum Common Subgraph among an indefinite number of graphs. We first extend a state-of-the-art branch-and-bound procedure working on two graphs to N graphs. Then, given the high computational cost of this approach, we trade off complexity for accuracy, and we propose a set of heuristics to approximate the exact solution for N graphs. We analyze sequential, parallel multi-core, and parallel-many core (GPU-based) approaches, exploiting several leveraging techniques to decrease the contention among threads, improve the workload balance of the different tasks, reduce the computation time, and increase the final result size. We also present several sorting heuristics to order the vertices of the graphs and the graphs themselves. We compare our algorithms with a state-of-the-art method on publicly available benchmark sets. On graph pairs, we are able to speed up the exact computation by a 2× factor, pruning the search space by more than 60%. On sets of more than two graphs, all exact solutions are extremely time-consuming and of a complex application in many real cases. On the contrary, our heuristics are far less expensive (as they show a lower-bound for the speed up of 10×), have a far better asymptotic complexity (with speed ups up to several orders of magnitude in our experiments), and obtain excellent approximations of the maximal solution with 98.5% of the nodes on average.

Homomorphism complexes, reconfiguration, and homotopy for directed graphs

The neighborhood complex of a graph was introduced by Lovász to provide topological lower bounds on chromatic number. More general homomorphism complexes of graphs were further studied by Babson and Kozlov. Such ‘Hom complexes’ are also related to mixings of graph colorings and other reconfiguration problems, as well as a notion of discrete homotopy for graphs. Here we initiate the detailed study of Hom complexes for directed graphs (digraphs). For any pair of digraphs graphs G and H, we consider the polyhedral complex Hom⃗(G,H) that parametrizes the directed graph homomorphisms f:G→H. This construction can be seen as a special case of the poset structure on the set of multihomomorphisms in more general categories, as introduced by Kozlov, Matsushita, and others. Hom complexes of digraphs have applications in the study of chains in graded posets and cellular resolutions of monomial ideals.We study examples of directed Hom⃗ complexes and relate their topological properties to certain graph operations including products, adjunctions, and foldings. We introduce a notion of a neighborhood complex for a digraph and prove that its homotopy type is recovered as the Hom⃗ complex of homomorphisms from a directed edge. We establish a number of results regarding the topology of directed neighborhood complexes, including the dependence on directed bipartite subgraphs, a digraph version of the Mycielski construction, as well as vanishing theorems for higher homology. The Hom⃗ complexes of digraphs provide a natural framework for reconfiguration of homomorphisms of digraphs. Inspired by notions of directed graph colorings we study the connectivity of Hom⃗(G,Tn) for Tn a tournament, obtaining a complete answer for the case of transitive Tn. If G=Tm is also a transitive tournament, we describe a connection to mixed subdivisions of dilated simplices. Finally we use paths in the internal hom objects of digraphs to define various notions of homotopy, and discuss connections to the topology of Hom⃗ complexes.

A Non-aligning Variant of Generalized Turán Problems

AbstractIn the so-called generalized Turán problems we study the largest number of copies of H in an n-vertex F-free graph G. Here we introduce a variant, where F is not forbidden, but we restrict how copies of H and F can be placed in G. More precisely, given an integer n and graphs H and F, what is the largest number of copies of H in an n-vertex graph such that the vertex set of that copy does not contain and is not contained in the vertex set of a copy of F? We solve this problem for some instances, give bounds in other instances, and we use our results to determine the generalized Turán number for some pairs of graphs.

Several Methods for Generating Families of Orderenergetic, Integral and Equienergetic Graphs

We define a general unary graph operation and give several applications of these operation in this paper. The adjacency matrix and the complete spectrum of the derived graphs are determined. Different methods for generating sequences of orderenergetic graphs from known orderenergetic graphs are described. Several methods are described for generating orderenergetic graphs from non-orderenergetic graphs. Methods for generating new families of integral graphs using this new operation are also discussed. It is also possible to generate infinite sequences of pair of equienergetic and non-cospectral graphs using this graph operation.

Kazhdan constants and isomorphic graph pairs

Let G be a finite group, and let Γ be a subset of G. The Kazhdan constant of the pair (G,Γ) is defined to bethe maximum distance we can guarantee that an arbitrary unitvector in an arbitrary nontrivial irreducible unitary representation space of G can be moved by some element of Γ. The Kazhdanconstant relates to the expansion properties of the Cayley graph generated by G and Γ, and has been much studied in this context. Different pairs (G1,Γ1) and (G2,Γ2) may give rise to isomorphic Cayley graphs. In this paper, we investigate the question: To whatextent is the Kazhdan constant a graph invariant? In other words, if the pairs yield isomorphic Cayley graphs, must the corresponding Kazhdan constants be equal? In our main theorem, we constructan infinite family of such pairs where the Kazhdan constants areunequal. Other relevant results are presented as well.

Generalized Paley graphs equienergetic with their complements

We consider generalized Paley graphs , generalized Paley sum graphs , and their corresponding complements and , for k = 3, 4. Denote by either or . We compute the spectra of and and from them we obtain the spectra of and also. Then we show that, in the non-semiprimitive case, the spectrum of and with p prime can be recursively obtained, under certain arithmetic conditions, from the spectrum of the graphs and for any , respectively. Using the spectra of these graphs we give necessary and sufficient conditions on the spectrum of such that and are equienergetic for k = 3, 4. In a previous work we have classified all bipartite regular graphs and all strongly regular graphs which are complementary equienergetic, i.e. and are equienergetic pairs of graphs. Here we construct infinite pairs of equienergetic non-isospectral regular graphs which are neither bipartite nor strongly regular.

On Independent Domination in Direct Products

In [16] Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that \(i(G\times H) \ge i(G)i(H)\) where i(G) is the independent domination number of G and \(G\times H\) is the direct product of graphs G and H. We show this conjecture is false, and, in fact, construct pairs of graphs for which \(\min \{i(G), i(H)\} - i(G\times H)\) is arbitrarily large. We also give the exact value of \(i(G\times K_n)\) when G is either a path or a cycle.

Type synthesis of deployable and symmetrical single-loop mechanisms for constructing aerospace platforms

This paper presents a general synthesis method for 1-DOF single-loop mechanisms (SLMs). This method divides an SLM into two portions: one is a revolute pair and another is a branch chain consisting of several kinematic pairs. The core idea is to design several kinematic pairs in the branch chain without constraining the rotation of the revolute pair. First, the constraint line graph of this revolute pair is drawn. The constraints of the branch chain are classified according to the equivalent constraint line graph of this revolute pair. Subsequently, the synthesis schemes corresponding to these constraints are formulated. Deployable and symmetrical SLMs are constructed, which include not only novel SLMs such as RS1P1RS3RS2P2R, and Rc1Rc2PRc2′Rc1′, but also classical SLMs such as Bennett and Bricard SLMs. It verifies the feasibility and effectiveness of the method. Finally, based on the overconstrained SLMs, another approach for obtaining more deployable non-overconstrained SLMs is proposed. By this approach, some non-overconstrained 7R SLMs are also obtained. Three examples are given to illustrate the potential applications of the obtained SLMs.

The Complexity of Two Colouring Games

We consider two variants of orthogonal colouring games on graphs. In these games, two players alternate colouring uncoloured vertices (from a choice of min {mathbb {N}} colours) of a pair of isomorphic graphs while respecting the properness and the orthogonality of the partial colourings. In the normal play variant, the first player unable to move loses. In the scoring variant, each player aims to maximise their score, which is the number of coloured vertices in their copy of the graph. We prove that, given an instance with partial colourings, both the normal play and the scoring variant of the game are PSPACE-complete. An involution sigma of a graph G is strictly matched if its fixed point set induces a clique and vsigma (v)in E(G) for any non-fixed point vin V(G). Andres et al. (Theor Comput Sci 795:312–325, 2019) gave a solution of the normal play variant played on graphs that admit a strictly matched involution. We prove that recognising graphs that admit a strictly matched involution is NP-complete.

Towards the 0-statement of the Kohayakawa-Kreuter conjecture

AbstractIn this paper, we study asymmetric Ramsey properties of the random graph $G_{n,p}$ . Let $r \in \mathbb{N}$ and $H_1, \ldots, H_r$ be graphs. We write $G_{n,p} \to (H_1, \ldots, H_r)$ to denote the property that whenever we colour the edges of $G_{n,p}$ with colours from the set $[r] \,{:\!=}\, \{1, \ldots, r\}$ there exists $i \in [r]$ and a copy of $H_i$ in $G_{n,p}$ monochromatic in colour $i$ . There has been much interest in determining the asymptotic threshold function for this property. In several papers, Rödl and Ruciński determined a threshold function for the general symmetric case; that is, when $H_1 = \cdots = H_r$ . A conjecture of Kohayakawa and Kreuter from 1997, if true, would fully resolve the asymmetric problem. Recently, the $1$ -statement of this conjecture was confirmed by Mousset, Nenadov and Samotij.Building on work of Marciniszyn, Skokan, Spöhel and Steger from 2009, we reduce the $0$ -statement of Kohayakawa and Kreuter’s conjecture to a certain deterministic subproblem. To demonstrate the potential of this approach, we show this subproblem can be resolved for almost all pairs of regular graphs. This therefore resolves the $0$ -statement for all such pairs of graphs.

ON THE COMPLETE PRODUCT OF FUZZY GRAPHS

Strong fuzzy graph and complete fuzzy graph are two special kinds of fuzzy graphs in which each edge membership value equals the least of its vertex membership values incidental to the edge. In this research study, we obtain some interesting results on the complete product of a pair of strong fuzzy graphs as well as the complete product of a pair of complete fuzzy graphs. In precise, we prove that the complete product of two strong fuzzy graphs is again a strong fuzzy graph and the complete product of two complete fuzzy graphs is again a complete fuzzy graph. Also, we discuss the conditions under which the property of regularity will be mutually transmitted between the complete product of two fuzzy graphs and one of its factor fuzzy graphs

An Information-Theoretic Bound on p-Values for Detecting Communities Shared between Weighted Labeled Graphs

Extraction of subsets of highly connected nodes (“communities” or modules) is a standard step in the analysis of complex social and biological networks. We here consider the problem of finding a relatively small set of nodes in two labeled weighted graphs that is highly connected in both. While many scoring functions and algorithms tackle the problem, the typically high computational cost of permutation testing required to establish the p-value for the observed pattern presents a major practical obstacle. To address this problem, we here extend the recently proposed CTD (“Connect the Dots”) approach to establish information-theoretic upper bounds on the p-values and lower bounds on the size and connectedness of communities that are detectable. This is an innovation on the applicability of CTD, broadening its use to pairs of graphs.

A Graph Matching Model for Designer Team Selection for Collaborative Design Crowdsourcing Tasks in Social Manufacturing

In order to find a suitable designer team for the collaborative design crowdsourcing task of a product, we consider the matching problem between collaborative design crowdsourcing task network graph and the designer network graph. Due to the difference in the nodes and edges of the two types of graphs, we propose a graph matching model based on a similar structure. The model first uses the Graph Convolutional Network to extract features of the graph structure to obtain the node-level embeddings. Secondly, an attention mechanism considering the differences in the importance of different nodes in the graph assigns different weights to different nodes to aggregate node-level embeddings into graph-level embeddings. Finally, the graph-level embeddings of the two graphs to be matched are input into a multi-layer fully connected neural network to obtain the similarity score of the graph pair after they are obtained from the concat operation. We compare our model with the basic model based on four evaluation metrics in two datasets. The experimental results show that our model can more accurately find graph pairs based on a similar structure. The crankshaft linkage mechanism produced by the enterprise is taken as an example to verify the practicality and applicability of our model and method.

Spectra of Complemented Triangulation Graphs

The complemented triangulation graph of a graph G, denoted by CT(G), is defined as the graph obtained from G by adding, for each edge uv of G, a new vertex whose neighbours are the vertices of G other than u and v. In this paper, we first obtain the A-spectra, the L-spectra, and the Q-spectra of the complemented triangulation graphs of regular graphs. By using the results, we construct infinitely many pairs of A-cospectral graphs, L-cospectral graphs, and Q-cospectral graphs. We also obtain the number of spanning trees and the Kirchhoff index of the complemented triangulation graphs of regular graphs.

ISI-Equienergetic Graphs

The ISI-energy εisi(G) of a graph G=(V,E) is the sum of the absolute values of the eigenvalues of the ISI-matrix C(G)=[cij]n×n in which cij=d(vi)d(vj)d(vi)+d(vj) if vivj∈E(G) and cij=0 otherwise. d(vi) denotes the degree of vertex vi∈V. As a class of graph energy, ISI-energy can be utilized to ascertain the general energy of conjugated carbon molecules. Two non-isomorphic graphs of the same order are said to be ISI-equienergetic if their ISI-energies are equal. In this paper, we construct pairs of connected, ISI-noncospectral, ISI-equienergetic graphs of order n for all n≥9. In addition, for n-vertex r(r≥3)-regular graph G, and for each k≥2, we obtain εisi(Lk(G)¯), depending only on n and r. This result enables a systematic construction of pairs of ISI-noncospectral graphs of the same order, having equal ISI-energies.

Energy and Randić energy of special graphs

In this paper, we determine the Randić energy of the m-splitting graph, the m-shadow graph and the m-duplicate graph of a given graph, m being an arbitrary integer. Our results allow the construction of an infinite sequence of graphs having the same Randić energy. Further, we determine some graph invariants like the degree Kirchhoff index, the Kemeny’s constant and the number of spanning trees of some special graphs. From our results, we indicate how to obtain infinitely many pairs of equienergetic graphs, Randić equienergetic graphs and also, infinite families of integral graphs.

Asymmetric Ramsey properties of random graphs involving cliques and cycles

AbstractWe say that if, in every edge coloring , we can find either a 1‐colored copy of or a 2‐colored copy of . The well‐known Kohayakawa‐Kreuter conjecture states that the threshold for the property is equal to , where is given by for any pair of graphs and with . In this article, we show the 0‐statement of the Kohayakawa–Kreuter conjecture for every pair of cycles and cliques.

Synthesis for observability of logical control networks

Finite-state systems have applications in systems biology, formal verification and synthesis of infinite-state (hybrid) systems, etc. As deterministic finite-state systems, logical control networks (LCNs) consist of a finite number of nodes which can be in a finite number of states and update their states. In this paper, we investigate the synthesis problem for observability of LCNs based on state feedback with exogenous input by using the semitensor product proposed by Daizhan Cheng and the notion of observability graph (previously called weighted pair graph) proposed by us. We prove that if an unobservable LCN can be made observable by state feedback with exogenous input, then it can also be made observable by state feedback (without exogenous input, equivalent to state feedback with constant input). Furthermore, we give an upper bound on the number of state-feedback controllers that are needed to be tested in order to verify whether an unobservable LCN can be made observable by state feedback, and based on the procedure of obtaining the upper bound, we design an observability synthesis algorithm, by additionally combining the ideas of a greedy algorithm and dynamic programming. These results open the study of observability synthesis in LCNs.

The geometry of Gaussian double Markovian distributions

AbstractGaussian double Markovian models consist of covariance matrices constrained by a pair of graphs specifying zeros simultaneously in the matrix and its inverse. We study the semi‐algebraic geometry of these models, in particular their dimension, smoothness, and connectedness as well as algebraic and combinatorial properties.

Interpretable Neural Subgraph Matching for Graph Retrieval

Given a query graph and a database of corpus graphs, a graph retrieval system aims to deliver the most relevant corpus graphs. Graph retrieval based on subgraph matching has a wide variety of applications, e.g., molecular fingerprint detection, circuit design, software analysis, and question answering. In such applications, a corpus graph is relevant to a query graph, if the query graph is (perfectly or approximately) a subgraph of the corpus graph. Existing neural graph retrieval models compare the node or graph embeddings of the query-corpus pairs, to compute the relevance scores between them. However, such models may not provide edge consistency between the query and corpus graphs. Moreover, they predominantly use symmetric relevance scores, which are not appropriate in the context of subgraph matching, since the underlying relevance score in subgraph search should be measured using the partial order induced by subgraph-supergraph relationship. Consequently, they show poor retrieval performance in the context of subgraph matching. In response, we propose ISONET, a novel interpretable neural edge alignment formulation, which is better able to learn the edge-consistent mapping necessary for subgraph matching. ISONET incorporates a new scoring mechanism which enforces an asymmetric relevance score, specifically tailored to subgraph matching. ISONET’s design enables it to directly identify the underlying subgraph in a corpus graph, which is relevant to the given query graph. Our experiments on diverse datasets show that ISONET outperforms recent graph retrieval formulations and systems. Additionally, ISONET can provide interpretable alignments between query-corpus graph pairs during inference, despite being trained only using binary relevance labels of whole graphs during training, without any fine-grained ground truth information about node or edge alignments.