Search results for: Detour eigenvalue (of a graph)

Authors: S.K.Ayyaswamy, S.Balachandran

Abstract:

The Detour matrix (DD) of a graph has for its ( i , j) entry the length of the longest path between vertices i and j. The DD-eigenvalues of a connected graph G are the eigenvalues for its detour matrix, and they form the DD-spectrum of G. The DD-energy EDD of the graph G is the sum of the absolute values of its DDeigenvalues. Two connected graphs are said to be DD- equienergetic if they have equal DD-energies. In this paper, the DD- spectra of a variety of graphs and their DD-energies are calculated.

Keywords: Detour eigenvalue (of a graph), detour spectrum(of a graph), detour energy(of a graph), detour - equienergetic graphs.

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Authors: Khidir R. Sharaf, Didar A. Ali

Abstract:

The nullity η(G) of a graph is the occurrence of zero as an eigenvalue in its spectra. A zero-sum weighting of a graph G is real valued function, say f from vertices of G to the set of real numbers, provided that for each vertex of G the summation of the weights f(w) over all neighborhood w of v is zero for each v in G.A high zero-sum weighting of G is one that uses maximum number of non-zero independent variables. If G is graph with an end vertex, and if H is an induced subgraph of G obtained by deleting this vertex together with the vertex adjacent to it, then, η(G)= η(H). In this paper, a high zero-sum weighting technique and the endvertex procedure are applied to evaluate the nullity of t-tupple and generalized t-tupple graphs are derived  and determined for some special types of graphs,

 Also, we introduce and prove some important results about the t-tupple coalescence, Cartesian and Kronecker products of nut graphs.

Keywords: Graph theory, Graph spectra, Nullity of graphs.

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Authors: S.K.Ayyaswamy, S.Balachandran, K.Kannan

Abstract:

Let G be a graph of order n. The second stage adjacency matrix of G is the symmetric n × n matrix for which the ijth entry is 1 if the vertices vi and vj are of distance two; otherwise 0. The sum of the absolute values of this second stage adjacency matrix is called the second stage energy of G. In this paper we investigate a few properties and determine some upper bounds for the largest eigenvalue.

Keywords: Second stage spectral radius, Irreducible matrix, Derived graph

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Authors: A. M. Nazari, B. Sepehrian, M. Jabari

Abstract:

In this paper we study the inverse eigenvalue problem for symmetric special matrices and introduce sufficient conditions for obtaining nonnegative matrices. We get the HROU algorithm from [1] and introduce some extension of this algorithm. If we have some eigenvectors and associated eigenvalues of a matrix, then by this extension we can find the symmetric matrix that its eigenvalue and eigenvectors are given. At last we study the special cases and get some remarkable results.

Keywords: Householder matrix, nonnegative matrix, Inverse eigenvalue problem.

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Authors: Nileshkumar Vaishnav, Aditya Tatu

Abstract:

An algebraic framework for processing graph signals axiomatically designates the graph adjacency matrix as the shift operator. In this setup, we often encounter a problem wherein we know the filtered output and the filter coefficients, and need to find out the input graph signal. Solution to this problem using direct approach requires O(N3) operations, where N is the number of vertices in graph. In this paper, we adapt the spectral graph partitioning method for partitioning of graphs and use it to reduce the computational cost of the filtering problem. We use the example of denoising of the temperature data to illustrate the efficacy of the approach.

Keywords: Graph signal processing, graph partitioning, inverse filtering on graphs, algebraic signal processing.

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Authors: Do Phuc, Nguyen Thi Kim Phung

Abstract:

In this paper, we represent protein structure by using graph. A protein structure database will become a graph database. Each graph is represented by a spectral vector. We use Jacobi rotation algorithm to calculate the eigenvalues of the normalized Laplacian representation of adjacency matrix of graph. To measure the similarity between two graphs, we calculate the Euclidean distance between two graph spectral vectors. To cluster the graphs, we use M-tree with the Euclidean distance to cluster spectral vectors. Besides, M-tree can be used for graph searching in graph database. Our proposal method was tested with graph database of 100 graphs representing 100 protein structures downloaded from Protein Data Bank (PDB) and we compare the result with the SCOP hierarchical structure.

Keywords: Eigenvalues, m-tree, graph database, protein structure, spectra graph theory.

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Authors: Hassan Manouzi, Taous-Meriem Laleg-Kirati

Abstract:

In this paper we study mathematically the eigenvalue problem for stochastic elliptic partial differential equation of Wick type. Using the Wick-product and the Wiener-Itô chaos expansion, the stochastic eigenvalue problem is reformulated as a system of an eigenvalue problem for a deterministic partial differential equation and elliptic partial differential equations by using the Fredholm alternative. To reduce the computational complexity of this system, we shall use a decomposition method using the Wiener-Itô chaos expansion. Once the approximation of the solution is performed using the finite element method for example, the statistics of the numerical solution can be easily evaluated.

Keywords: Eigenvalue problem, Wick product, SPDEs, finite element, Wiener-Itô chaos expansion.

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Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Abstract–Let k ≥ 3 be an integer, and let G be a graph of order n with n ≥ 9k +3- 42(k - 1)2 + 2. Then a spanning subgraph F of G is called a k-factor if dF (x) = k for each x ∈ V (G). A fractional k-factor is a way of assigning weights to the edges of a graph G (with all weights between 0 and 1) such that for each vertex the sum of the weights of the edges incident with that vertex is k. A graph G is a fractional k-deleted graph if there exists a fractional k-factor after deleting any edge of G. In this paper, it is proved that G is a fractional k-deleted graph if G satisfies δ(G) ≥ k + 1 and |NG(x) ∪ NG(y)| ≥ 1 2 (n + k - 2) for each pair of nonadjacent vertices x, y of G.

Keywords: Graph, minimum degree, neighborhood union, fractional k-factor, fractional k-deleted graph.

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Authors: M. S. Haghighat, A. Dolati, M. Tabari, E. Mohseni

Abstract:

The Merrifield-Simmons index of a graph G is defined as the total number of its independent sets. A (n, n + 2)-graph is a connected simple graph with n vertices and n + 2 edges. In this paper we characterize the (n, n+2)-graph with the largest Merrifield- Simmons index. We show that its Merrifield-Simmons index i.e. the upper bound of the Merrifield-Simmons index of the (n, n+2)-graphs is 9 × 2n-5 +1 for n ≥ 5.

Keywords: Merrifield-Simmons index, (n, n+2)-graph.

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Authors: R. Anitha, R. S. Lekshmi

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Graph decompositions are vital in the study of combinatorial design theory. A decomposition of a graph G is a partition of its edge set. An n-sun graph is a cycle Cn with an edge terminating in a vertex of degree one attached to each vertex. In this paper, we define n-sun decomposition of some even order graphs with a perfect matching. We have proved that the complete graph K2n, complete bipartite graph K2n, 2n and the Harary graph H4, 2n have n-sun decompositions. A labeling scheme is used to construct the n-suns.

Keywords: Decomposition, Hamilton cycle, n-sun graph, perfect matching, spanning tree.

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Authors: Shazia Javed, Noor Atinah Ahmad

Abstract:

The householder RLS (HRLS) algorithm is an O(N2) algorithm which recursively updates an arbitrary square-root of the input data correlation matrix and naturally provides the LS weight vector. A data dependent householder matrix is applied for such an update. In this paper a recursive estimate of the eigenvalue spread and misalignment of the algorithm is presented at a very low computational cost. Misalignment is found to be highly sensitive to the eigenvalue spread of input signals, output noise of the system and exponential window. Simulation results show noticeable degradation in the misalignment by increase in eigenvalue spread as well as system-s output noise, while exponential window was kept constant.

Keywords: HRLS algorithm, eigenvalue spread, misalignment.

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Authors: Tarasankar Pramanik, Sukumar Mondal, Madhumangal Pal

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In an interval graph G = (V,E) the distance between two vertices u, v is de£ned as the smallest number of edges in a path joining u and v. The eccentricity of a vertex v is the maximum among distances from all other vertices of V . The diameter (δ) and radius (ρ) of the graph G is respectively the maximum and minimum among all the eccentricities of G. The center of the graph G is the set C(G) of vertices with eccentricity ρ. In this context our aim is to establish the relation ρ = δ 2  for an interval graph and to determine the center of it.

Keywords: Interval graph, interval tree, radius, center.

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Authors: Sudhakar G

Abstract:

In this paper a new concept of partial complement of a graph G is introduced and using the same a new graph parameter, called completion number of a graph G, denoted by c(G) is defined. Some basic properties of graph parameter, completion number, are studied and upperbounds for completion number of classes of graphs are obtained , the paper includes the characterization also.

Keywords: Completion Number, Maximum Independent subset, Partial complements, Partial self complementary

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Authors: Sizhong Zhou, Hongxia Liu

Abstract:

Let G be a graph of order n, and let k  2 and m  0 be two integers. Let h : E(G)  [0, 1] be a function. If e∋x h(e) = k holds for each x  V (G), then we call G[Fh] a fractional k-factor of G with indicator function h where Fh = {e  E(G) : h(e) > 0}. A graph G is called a fractional (k,m)-deleted graph if there exists a fractional k-factor G[Fh] of G with indicator function h such that h(e) = 0 for any e  E(H), where H is any subgraph of G with m edges. In this paper, it is proved that G is a fractional (k,m)-deleted graph if (G)  k + m + m k+1 , n  4k2 + 2k − 6 + (4k 2 +6k−2)m−2 k−1 and max{dG(x), dG(y)}  n 2 for any vertices x and y of G with dG(x, y) = 2. Furthermore, it is shown that the result in this paper is best possible in some sense.

Keywords: Graph, degree condition, fractional k-factor, fractional (k, m)-deleted graph.

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Authors: M. Hussain, Aqsa Farooq

Abstract:

Let G = (V,E) be a connected graph and distance between any two vertices a and b in G is a−b geodesic and is denoted by d(a, b). A set of vertices W resolves a graph G if each vertex is uniquely determined by its vector of distances to the vertices in W. A metric dimension of G is the minimum cardinality of a resolving set of G. In this paper line graph of honeycomb network has been derived and then we calculated the metric dimension on line graph of honeycomb network.

Keywords: Resolving set, metric dimension, honeycomb network, line graph.

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Authors: Zdeňka Dostálová, David Matoušek, Bohumil Brtnik

Abstract:

As there are also graph methods of circuit analysis in addition to algebraic methods, it is, in theory, clearly possible to carry out an analysis of a whole switched circuit in two-phase switching exclusively by the graph method as well. This article deals with two methods of full-graph solving of switched circuits: by transformation graphs and by two-graphs. It deals with the circuit switched capacitors and the switched current, too. All methods are presented in an equally detailed steps to be able to compare.

Keywords: Switched capacitors of two phases, switched currents of two phases, transformation graph, two-graph, Mason's formula, voltage transfer, summary graph.

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Authors: Qiuyi Lyu, Bin Gong

Abstract:

Breadth-First Search (BFS) is a core graph algorithm that is widely used for graph analysis. As it is frequently used in many graph applications, improving the BFS performance is essential. In this paper, we present a graph ordering method that could reorder the graph nodes to achieve better data locality, thus, improving the BFS performance. Our method is based on an observation that the sibling relationships will dominate the cache access pattern during the BFS traversal. Therefore, we propose a frequency-based model to construct the graph order. First, we optimize the graph order according to the nodes’ visit frequency. Nodes with high visit frequency will be processed in priority. Second, we try to maximize the child nodes’ overlap layer by layer. As it is proved to be NP-hard, we propose a heuristic method that could greatly reduce the preprocessing overheads.We conduct extensive experiments on 16 real-world datasets. The result shows that our method could achieve comparable performance with the state-of-the-art methods while the graph ordering overheads are only about 1/15.

Keywords: Breadth-first search, BFS, graph ordering, graph algorithm.

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Authors: Israel Núñez-Hernández, Peter C. Breedveld, Paul B. T. Weustink, Gilberto Gonzalez-A

Abstract:

This paper proposes a phasor representation of electrical networks by using bond graph methodology. A so-called phasor bond graph is built up by means of two-dimensional bonds, which represent the complex plane. Impedances or admittances are used instead of the standard bond graph elements. A procedure to obtain the steady-state values from a phasor bond graph model is presented. Besides the presentation of a phasor bond graph library in SIDOPS code, also an application example is discussed.

Keywords: Bond graphs, phasor theory, steady-state, complex power, electrical networks.

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Authors: Hongzhi Wang, Jianzhong Li, Jizhou Luo

Abstract:

In many applications, data is in graph structure, which can be naturally represented as graph-structured XML. Existing queries defined on tree-structured and graph-structured XML data mainly focus on subgraph matching, which can not cover all the requirements of querying on graph. In this paper, a new kind of queries, topological query on graph-structured XML is presented. This kind of queries consider not only the structure of subgraph but also the topological relationship between subgraphs. With existing subgraph query processing algorithms, efficient algorithms for topological query processing are designed. Experimental results show the efficiency of implementation algorithms.

Keywords: XML, Graph Structure, Topological query.

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Authors: Lv Yuhua

Abstract:

In this paper, by constructing a special cone and using fixed point theorem and fixed point index theorem of cone, we get the existence of positive solution for a class of singular eigenvalue value problems with p-Laplace operator, which improved and generalized the result of related paper.

Keywords: Cone, fixed point index, eigenvalue problem, p-Laplace operator, positive solutions.

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Authors: Xiantong Li, Jianzhong Li

Abstract:

Graph has become increasingly important in modeling complicated structures and schemaless data such as proteins, chemical compounds, and XML documents. Given a graph query, it is desirable to retrieve graphs quickly from a large database via graph-based indices. Different from the existing methods, our approach, called VFM (Vertex to Frequent Feature Mapping), makes use of vertices and decision features as the basic indexing feature. VFM constructs two mappings between vertices and frequent features to answer graph queries. The VFM approach not only provides an elegant solution to the graph indexing problem, but also demonstrates how database indexing and query processing can benefit from data mining, especially frequent pattern mining. The results show that the proposed method not only avoids the enumeration method of getting subgraphs of query graph, but also effectively reduces the subgraph isomorphism tests between the query graph and graphs in candidate answer set in verification stage.

Keywords: Decision Feature, Frequent Feature, Graph Dataset, Graph Query

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Authors: Sizhong Zhou, Yang Xu

Abstract:

A graph G is fractional k-covered if for each edge e of G, there exists a fractional k-factor h, such that h(e) = 1. If k = 2, then a fractional k-covered graph is called a fractional 2-covered graph. The binding number bind(G) is defined as follows, bind(G) = min{|NG(X)| |X| : ├ÿ = X Ôèå V (G),NG(X) = V (G)}. In this paper, it is proved that G is fractional 2-covered if δ(G) ≥ 4 and bind(G) > 5 3 .

Keywords: graph, binding number, fractional k-factor, fractional k-covered graph.

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Authors: Benshi Zhu

Abstract:

In this paper, multiple positive solutions for semipositone discrete eigenvalue problems are obtained by using a three critical points theorem for nondifferentiable functional.

Keywords: Discrete eigenvalue problems, positive solutions, semipositone, three critical points theorem

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Authors: Marek Skomorowski

Abstract:

In syntactic pattern recognition a pattern can be represented by a graph. Given an unknown pattern represented by a graph g, the problem of recognition is to determine if the graph g belongs to a language L(G) generated by a graph grammar G. The so-called IE graphs have been defined in [1] for a description of patterns. The IE graphs are generated by so-called ETPL(k) graph grammars defined in [1]. An efficient, parsing algorithm for ETPL(k) graph grammars for syntactic recognition of patterns represented by IE graphs has been presented in [1]. In practice, structural descriptions may contain pattern distortions, so that the assignment of a graph g, representing an unknown pattern, to a graph language L(G) generated by an ETPL(k) graph grammar G is rejected by the ETPL(k) type parsing. Therefore, there is a need for constructing effective parsing algorithms for recognition of distorted patterns. The purpose of this paper is to present a new approach to syntactic recognition of distorted patterns. To take into account all variations of a distorted pattern under study, a probabilistic description of the pattern is needed. A random IE graph approach is proposed here for such a description ([2]).

Keywords: Syntactic pattern recognition, Distorted patterns, Random graphs, Graph grammars.

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Authors: Mana Tarjoman, Shaghayegh Zarei

Abstract:

Using efficient classification methods is necessary for automatic fingerprint recognition system. This paper introduces a new structural approach to fingerprint classification by using the directional image of fingerprints to increase the number of subclasses. In this method, the directional image of fingerprints is segmented into regions consisting of pixels with the same direction. Afterwards the relational graph to the segmented image is constructed and according to it, the super graph including prominent information of this graph is formed. Ultimately we apply a matching technique to compare obtained graph with the model graphs in order to classify fingerprints by using cost function. Increasing the number of subclasses with acceptable accuracy in classification and faster processing in fingerprints recognition, makes this system superior.

Keywords: Classification, Directional image, Fingerprint, Graph, Super graph.

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Authors: Sudhakara G, Hemanth Kumar A.R

Abstract:

In this paper, we define distance partition of vertex set of a graph G with reference to a vertex in it and with the help of the same, a graph with metric dimension two (i.e. β (G) = 2 ) is characterized. In the process, we develop a polynomial time algorithm that verifies if the metric dimension of a given graph G is two. The same algorithm explores all metric bases of graph G whenever β (G) = 2 . We also find a bound for cardinality of any distance partite set with reference to a given vertex, when ever β (G) = 2 . Also, in a graph G with β (G) = 2 , a bound for cardinality of any distance partite set as well as a bound for number of vertices in any sub graph H of G is obtained in terms of diam H .

Keywords: Metric basis, Distance partition, Metric dimension.

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Authors: Jing Li, Guang Zhou

Abstract:

Let A and B be nonnegative matrices. A new upper bound on the spectral radius ρ(A◦B) is obtained. Meanwhile, a new lower bound on the smallest eigenvalue q(AB) for the Fan product, and a new lower bound on the minimum eigenvalue q(B ◦A−1) for the Hadamard product of B and A−1 of two nonsingular M-matrices A and B are given. Some results of comparison are also given in theory. To illustrate our results, numerical examples are considered.

Keywords: Hadamard product, Fan product; nonnegative matrix, M-matrix, Spectral radius, Minimum eigenvalue, 1-path cover.

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Authors: Rajib Kumar Jha, P.K.Biswas, B.N.Chatterji

Abstract:

In this paper a new concept of partial complement of a graph G is introduced and using the same a new graph parameter, called completion number of a graph G, denoted by c(G) is defined. Some basic properties of graph parameter, completion number, are studied and upperbounds for completion number of classes of graphs are obtained , the paper includes the characterization also.

Keywords: Completion Number, Maximum Independent subset, Partial complements, Partial self complementary.

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Authors: Gilberto Gonzalez-A, Noe Barrera-G

Abstract:

An analysis of a synchronous generator in a bond graph approach is proposed. This bond graph allows to determine the simplified models of the system by using singular perturbations. Firstly, the nonlinear bond graph of the generator is linearized. Then, the slow and fast state equations by applying singular perturbations are obtained. Also, a bond graph to get the quasi-steady state of the slow dynamic is proposed. In order to verify the effectiveness of the singularly perturbed models, simulation results of the complete system and reduced models are shown.

Keywords: Bond graph modelling, synchronous generator, singular perturbations

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Authors: Priyadarsini P. L. K, Hemalatha T.

Abstract:

This paper proves that the problem of finding connected vertex cover in a 2-connected planar graph ( CVC-2 ) with maximum degree 4 is NP-complete. The motivation for proving this result is to give a shorter and simpler proof of NP-Completeness of TRA-MLC (the Top Right Access point Minimum-Length Corridor) problem [1], by finding the reduction from CVC-2. TRA-MLC has many applications in laying optical fibre cables for data communication and electrical wiring in floor plans.The problem of finding connected vertex cover in any planar graph ( CVC ) with maximum degree 4 is NP-complete [2]. We first show that CVC-2 belongs to NP and then we find a polynomial reduction from CVC to CVC-2. Let a graph G0 and an integer K form an instance of CVC, where G0 is a planar graph and K is an upper bound on the size of the connected vertex cover in G0. We construct a 2-connected planar graph, say G, by identifying the blocks and cut vertices of G0, and then finding the planar representation of all the blocks of G0, leading to a plane graph G1. We replace the cut vertices with cycles in such a way that the resultant graph G is a 2-connected planar graph with maximum degree 4. We consider L = K -2t+3 t i=1 di where t is the number of cut vertices in G1 and di is the number of blocks for which ith cut vertex is common. We prove that G will have a connected vertex cover with size less than or equal to L if and only if G0 has a connected vertex cover of size less than or equal to K.

Keywords: NP-complete, 2-Connected planar graph, block, cut vertex

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