Abstract: In an urban area the location allocation of emergency
services mobile units, such as ambulances, police patrol cars must be
designed so as to achieve a prompt response to demand locations.
In this paper the partition of a given urban network into distinct
sub-networks is performed such that the vertices in each component
are close and simultaneously the sums of the corresponding
population in the sub-networks are almost uniform. The objective
here is to position appropriately in each sub-network a mobile
emergency unit in order to reduce the response time to the demands.
A mathematical model in framework of graph theory is developed.
In order to clarify the corresponding method a relevant numerical
example is presented on a small network.
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.
Abstract: The topological distance between a pair of vertices i and j, which is denoted by d(vi, vj), is the number of edges of the shortest path joining i and j. The Wiener index W(G) is the sum of distances between all pairs of vertices of a graph G. W(G) = i
Abstract: The DNA microarray technology concurrently monitors the expression levels of thousands of genes during significant biological processes and across the related samples. The better understanding of functional genomics is obtained by extracting the patterns hidden in gene expression data. It is handled by clustering which reveals natural structures and identify interesting patterns in the underlying data. In the proposed work clustering gene expression data is done through an Advanced Nelder Mead (ANM) algorithm. Nelder Mead (NM) method is a method designed for optimization process. In Nelder Mead method, the vertices of a triangle are considered as the solutions. Many operations are performed on this triangle to obtain a better result. In the proposed work, the operations like reflection and expansion is eliminated and a new operation called spread-out is introduced. The spread-out operation will increase the global search area and thus provides a better result on optimization. The spread-out operation will give three points and the best among these three points will be used to replace the worst point. The experiment results are analyzed with optimization benchmark test functions and gene expression benchmark datasets. The results show that ANM outperforms NM in both benchmarks.
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.
Abstract: The unique structural configuration found in human foot allows easy walking. Similar movement is hard to imitate even for an ape. It is obvious that human ambulation relates to the foot structure itself. Suppose the bones are represented as vertices and the joints as edges. This leads to the development of a special graph that represents human foot. On a footprint there are point-ofcontacts which have contact with the ground. It involves specific vertices. Theoretically, for an ideal ambulation, these points provide reactions onto the ground or the static equilibrium forces. They are arranged in sequence in form of a path. The ambulating footprint follows this path. Having the human foot graph and the path crossbred, it results in a representation that describes the profile of an ideal ambulation. This profile cites the locations where the point-of-contact experience normal reaction forces. It highlights the significant of these points.
Abstract: In this paper, a method for matching image segments
using triangle-based (geometrical) regions is proposed. Triangular
regions are formed from triples of vertex points obtained from a
keypoint detector (SIFT). However, triangle regions are subject to
noise and distortion around the edges and vertices (especially acute
angles). Therefore, these triangles are expanded into parallelogramshaped
regions. The extracted image segments inherit an important
triangle property; the invariance to affine distortion. Given two
images, matching corresponding regions is conducted by computing
the relative affine matrix, rectifying one of the regions w.r.t. the other
one, then calculating the similarity between the reference and
rectified region. The experimental tests show the efficiency and
robustness of the proposed algorithm against geometrical distortion.
Abstract: Qk
n has been shown as an alternative to the hypercube
family. For any even integer k ≥ 4 and any integer n ≥ 2, Qk
n is
a bipartite graph. In this paper, we will prove that given any pair of
vertices, w and b, from different partite sets of Qk
n, there exist 2n
internally disjoint paths between w and b, denoted by {Pi | 0 ≤ i ≤ 2n-1}, such that 2n-1
i=0 Pi covers all vertices of Qk
n. The result is
optimal since each vertex of Qk
n has exactly 2n neighbors.
Abstract: In a graph G, a cycle is Hamiltonian cycle if it contain all vertices of G. Two Hamiltonian cycles C_1 = 〈u_0, u_1, u_2, ..., u_{n−1}, u_0〉 and C_2 = 〈v_0, v_1, v_2, ..., v_{n−1}, v_0〉 in G are independent if u_0 = v_0, u_i = ̸ v_i for all 1 ≤ i ≤ n−1. In G, a set of Hamiltonian cycles C = {C_1, C_2, ..., C_k} is mutually independent if any two Hamiltonian cycles of C are independent. The mutually independent Hamiltonicity IHC(G), = k means there exist a maximum integer k such that there exists k-mutually independent Hamiltonian cycles start from any vertex of G. In this paper, we prove that IHC(C_n × C_n) = 4, for n ≥ 3.
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 .
Abstract: In this paper a deterministic polynomial-time
algorithm is presented for the Clique problem. The case is considered
as the problem of omitting the minimum number of vertices from the
input graph so that none of the zeroes on the graph-s adjacency
matrix (except the main diagonal entries) would remain on the
adjacency matrix of the resulting subgraph. The existence of a
deterministic polynomial-time algorithm for the Clique problem, as
an NP-complete problem will prove the equality of P and NP
complexity classes.
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.
Abstract: Suppose G(V,E) is a graph, a function f : V \cup E \to \{1, 2, 3, \cdots, k\} is called the total edge(vertex) irregular k-labelling for G such that for each two edges are different having distinct weights. The total edge(vertex) irregularity strength of G, denoted by tes(G)(tvs(G), is the smallest k positive integers such that G has a total edge(vertex) irregular k-labelling. In this paper, we determined the total edge(vertex) irregularity strength of an amalgamation of two isomorphic cycles. The total edge irregularity strength and the total vertex irregularity strength of two isomorphic cycles on n vertices are \lceil (2n+2)/3 \rceil and \lceil 2n/3 \rceil for n \geq 3, respectively.
Abstract: The Maximum Weighted Independent Set (MWIS)
problem is a classic graph optimization NP-hard problem. Given an
undirected graph G = (V, E) and weighting function defined on the
vertex set, the MWIS problem is to find a vertex set S V whose total
weight is maximum subject to no two vertices in S are adjacent. This
paper presents a novel approach to approximate the MWIS of a graph
using minimum weighted vertex cover of the graph. Computational
experiments are designed and conducted to study the performance
of our proposed algorithm. Extensive simulation results show that
the proposed algorithm can yield better solutions than other existing
algorithms found in the literature for solving the MWIS.
Abstract: An induced acyclic graphoidal cover of a graph G is a
collection ψ of open paths in G such that every path in ψ has atleast
two vertices, every vertex of G is an internal vertex of at most one
path in ψ, every edge of G is in exactly one path in ψ and every
member of ψ is an induced path. The minimum cardinality of an
induced acyclic graphoidal cover of G is called the induced acyclic
graphoidal covering number of G and is denoted by ηia(G) or ηia.
Here we find induced acyclic graphoidal cover for some classes of
graphs.
Abstract: The classification of the protein structure is commonly
not performed for the whole protein but for structural domains, i.e.,
compact functional units preserved during evolution. Hence, a first
step to a protein structure classification is the separation of the
protein into its domains. We approach the problem of protein domain
identification by proposing a novel graph theoretical algorithm. We
represent the protein structure as an undirected, unweighted and
unlabeled graph which nodes correspond the secondary structure
elements of the protein. This graph is call the protein graph. The
domains are then identified as partitions of the graph corresponding
to vertices sets obtained by the maximization of an objective function,
which mutually maximizes the cycle distributions found in the
partitions of the graph. Our algorithm does not utilize any other kind
of information besides the cycle-distribution to find the partitions. If
a partition is found, the algorithm is iteratively applied to each of
the resulting subgraphs. As stop criterion, we calculate numerically
a significance level which indicates the stability of the predicted
partition against a random rewiring of the protein graph. Hence,
our algorithm terminates automatically its iterative application. We
present results for one and two domain proteins and compare our
results with the manually assigned domains by the SCOP database
and differences are discussed.
Abstract: An induced graphoidal cover of a graph G is a collection ψ of (not necessarily open) paths in G such that every path in ψ has at least two vertices, every vertex of G is an internal vertex of at most one path in ψ, every edge of G is in exactly one path in ψ and every member of ψ is an induced cycle or an induced path. The minimum cardinality of an induced graphoidal cover of G is called the induced graphoidal covering number of G and is denoted by ηi(G) or ηi. Here we find induced graphoidal cover for some classes of graphs.
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.
Abstract: The similarity comparison of RNA secondary
structures is important in studying the functions of RNAs. In recent
years, most existing tools represent the secondary structures by
tree-based presentation and calculate the similarity by tree alignment
distance. Different to previous approaches, we propose a new method
based on maximum clique detection algorithm to extract the maximum
common structural elements in compared RNA secondary structures.
A new graph-based similarity measurement and maximum common
subgraph detection procedures for comparing purely RNA secondary
structures is introduced. Given two RNA secondary structures, the
proposed algorithm consists of a process to determine the score of the
structural similarity, followed by comparing vertices labelling, the
labelled edges and the exact degree of each vertex. The proposed
algorithm also consists of a process to extract the common structural
elements between compared secondary structures based on a proposed
maximum clique detection of the problem. This graph-based model
also can work with NC-IUB code to perform the pattern-based
searching. Therefore, it can be used to identify functional RNA motifs
from database or to extract common substructures between complex
RNA secondary structures. We have proved the performance of this
proposed algorithm by experimental results. It provides a new idea of
comparing RNA secondary structures. This tool is helpful to those
who are interested in structural bioinformatics.
Abstract: The vertex connectivity of a graph is the smallest number of vertices whose deletion separates the graph or makes it trivial. This work is devoted to the problem of vertex connectivity test of graphs in a distributed environment based on a general and a constructive approach. The contribution of this paper is threefold. First, using a preconstructed spanning tree of the considered graph, we present a protocol to test whether a given graph is 2-connected using only local knowledge. Second, we present an encoding of this protocol using graph relabeling systems. The last contribution is the implementation of this protocol in the message passing model. For a given graph G, where M is the number of its edges, N the number of its nodes and Δ is its degree, our algorithms need the following requirements: The first one uses O(Δ×N2) steps and O(Δ×logΔ) bits per node. The second one uses O(Δ×N2) messages, O(N2) time and O(Δ × logΔ) bits per node. Furthermore, the studied network is semi-anonymous: Only the root of the pre-constructed spanning tree needs to be identified.