Two-connected Graphs Research Articles

Modification in Two-Connected Graph with Gallai’s Property in 2-Dimensional and 3-Dimensional Graph Containing 19 Vertices

The graph theory plays an important role in the network analysis, social networking as well as in many engineering fields such as electrical circuits, artificial intelligence, architecture, making the design or pattern of roads, buildings, shopping mall and etc. Due to this wide range application human enjoying her life with peacefully, Graph theory creates a way for human being to connect among themselves by social network. All above applications based on graph or molecule which may be the planer, non-planer and Peterson graph or etc. Peterson graph is the most important and reasonable example of Hypo-Hamiltonian graph. In the earlier, it was found as a hypo-traceable graph (graph which has not Hamiltonian graph. Naeem et al has worked on “A Two-Connected Graph with Gallai’s Property” In his research paper he has applied the property and has found the longest path and cycle in the graph. In this research paper we will develop the 3-dimensional graph of computational molecule contains 19 vertices and will split it into three different planes (xy, xz and yz-plane), and will find the longest path, longest cycle the molecule. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

A 2-connected graph in two dimensions containing 25 vertices with Gallai's Property

Graph theory recently considers as a modern field of mathematics and it was introduced by a great mathematician Leonhard Euler in 1735. Late then it is a flowered in the strong tool used in closely each area of science and nowadays it is most attractive and active area of mathematics research. Actually, graph theory is the study of graph, structure and molecules which have association in each other. It is very essential part of discrete mathematics which has been created with collaboration of nodes (vertices) and edges (lines or links), if a graph is connected with vertices directly by edges it is direct graph or a graph having symmetrically lines is also called direct graph or diagraph. In the studies of graph, we have investigated so many kinds of graph, Hypo-Hamiltonian is one of them. It is most important and it can be defined as the graph which hasn't Hamiltonian cycle but can be developed with the removing of single vertex form the developed graph. Lot of researchers have given the contribution in this research just like Naeem et al has worked on "A Two-Connected Graph with Gallai's Property" containing 12,18 and 25 vertices. In this research paper we have developed two different graphs consisting 25 Eulerian graph which has highest cycle and path order C( G) — 24 and P( G) — 25. Other graph is in 3D under cycle which contains 23 nodes.

Revisiting Groeneveld’s approach to the virial expansion

A generalized version of Groeneveld’s convergence criterion for the virial expansion and generating functionals for weighted two-connected graphs is proven. This criterion works for inhomogeneous systems and yields bounds for the density expansions of the correlation functions ρs (a.k.a. distribution functions or factorial moment measures) of grand-canonical Gibbs measures with pairwise interactions. The proof is based on recurrence relations for graph weights related to the Kirkwood–Salsburg integral equation for correlation functions. The proof does not use an inversion of the density-activity expansion; however, a Möbius inversion on the lattice of set partitions enters the derivation of the recurrence relations.

A Two-Connected Graph with Gallai’s Property

The most famous examples of Hypo-Hamiltonian graph is the Petersen graph. Before the discovery of Hypo-traceable graphs, Tibor Gallai, in 1966, raised the question whether the graphs in which each vertex is missed by some longest path. This property will be called Gallai’s property, various authors worked on that property. In 1969, Gallai’s question was first replied through H. Walther, who introduced a planar graph on 25 vertices satisfying Gallai’s criterion. Furthermore, H. Walther and H. Voss and Tudor Zamfirescu introduced the graph with 12 vertices and it was guessed that order 12 is the smaller possibility of such a graphLater the question was modifies by Tudor Zamfirescu and asked that whether there exists graphs of Paths and Cycles, that is to say i-connected graphs (planar or non-planar respectively), such that each set of j points are disjoint from some longest paths or cycles., Several good examples answering Tudor Zamfirescu’s questions were published. In this note a graphs is developed with the property that everyone vertex is missed by some longest cycle with connectivity 2, satisfying Gallai’s property. The designed graphs can be useful in various fields of science and technology including computational geometry, networking, theoretical computer science and circuit designing.

Convergence of Density Expansions of Correlation Functions and the Ornstein\u2013Zernike Equation

We prove absolute convergence of the multi-body correlation functions as a power series in the density uniformly in their arguments. This is done by working in the context of the cluster expansion in the canonical ensemble and by expressing the correlation functions as the derivative of the logarithm of an appropriately extended partition function. In the thermodynamic limit, due to combinatorial cancellations, we show that the coefficients of the above series are expressed by sums over some class of two-connected graphs. Furthermore, we prove the convergence of the density expansion of the “direct correlation function” which is based on a completely different approach and it is valid only for some integral norm. Precisely, this integral norm is suitable to derive the Ornstein–Zernike equation. As a further outcome, we obtain a rigorous quantification of the error in the Percus–Yevick approximation.

A solution to the combinatorial puzzle of Mayer’s virial expansion

Mayer’s second theorem in the context of a classical gasmodel allows us to write the coefficients of the virial expansion of pressure in terms of weighted two-connected graphs. Labelle, Leroux and Ducharme studied the graph weights arising from the one-dimensional hardcore gas model and noticed that the sum of these weights over all two-connected graphs with n vertices is –n(n–2)! . This paper addresses the question of achieving a purely combinatorial proof of this observation and extends the proof of Bernardi for the connected graph case.

Multispecies Virial Expansions

We study the virial expansion of mixtures of countably many different types of particles. The main tool is the Lagrange-Good inversion formula, which has other applications such as counting coloured trees or studying probability generating functions in multi-type branching processes. We prove that the virial expansion converges absolutely in a domain of small densities. In addition, we establish that the virial coefficients can be expressed in terms of two-connected graphs.

On the minimum number of edges of two-connected graphs with given diameter

We answer the following question: what is the minimum number of edges of a 2-connected graph with a given diameter? This problem stems from survivable telecommunication network design with grade-of-service constraints. In this paper, we prove tight bounds for 2-connected graphs and for 2-edge-connected graphs. The bound for 2-connected graphs was a conjecture of B. Bollobás (AMH 75–80) [3].

Structure and enumeration of two-connected graphs with prescribed three-connected components

We adapt the classical 3-decomposition of any 2-connected graph to the case of simple graphs (no loops or multiple edges). By analogy with the block-cutpoint tree of a connected graph, we deduce from this decomposition a bicolored tree tc ( g ) associated with any 2-connected graph g, whose white vertices are the 3 -components of g (3-connected components or polygons) and black vertices are bonds arising from separating pairs of vertices of g and linking together the 3-components. Two fundamental species relationships on graphs and networks follow from this construction. The first one is a dissymmetry theorem which leads to the expression of the class B = B ( F ) of 2-connected graphs, all of whose 3-connected components belong to a given class F of 3-connected graphs, in terms of various rootings of B . The second one is a functional equation which characterizes the corresponding class R = R ( F ) of two-pole networks all of whose 3-connected components are in F . All the rootings of B are then expressed in terms of F and R . There follow corresponding identities for all the associated power series, in particular, the edge index series. These results are expressed in terms of species of structures. Using these results we enumerate several classes of labelled and unlabelled graphs, including series-parallel graphs, 2-connected planar graphs, K 3 , 3 -free 2-connected graphs and K 3 , 3 -free projective planar and toroidal graphs.

Quality-of-service and quality-of-protection issues in preplanned recovery schemes using redundant trees

We study quality-of-service (QoS) and quality-of-protection (QoP) issues in redundant tree based preplanned recovery schemes for a single-link failure in two-edge connected graphs and for a single-node failure in two-connected graphs. We present schemes (to be called G-MFBG schemes) that generalize the schemes (to be called MFBG schemes) developed by Me/spl acute/dard et al. (1997) to construct a pair of redundant trees, called red and blue trees, which guarantees fast recovery from any single-link/node failure, as long as the failed node is not the root node. Using the G-MFBG schemes, we study QoS issues relating to red/blue trees. We present effective heuristics for computing a pair of redundant trees with low average delay or small total cost. We develop an optimal algorithm for computing a pair of red/blue trees with maximum bandwidth. Furthermore, a pair of red/blue trees guarantees fast recovery from simultaneous multiple failures if it satisfies certain properties. This leads us to define the concept of QoP of a pair of red/blue trees. We present an effective heuristic to construct a pair of red/blue trees with high QoP. The paper concludes with a discussion of computational results that demonstrate the effectiveness of the different algorithms presented.

The Bounded Cycle-Cover Problem

We consider the bounded cycle-cover problem, which is to find a minimum cost cycle cover of a two-connected graph such that no cycle in the cover contains more than a prescribed numbered of edges. This problem arises in the design of fiber-optic telecommunications networks that employ multiple self-healing rings to provide routing for communication traffic, even in the event of a fiber cut or other type of link failure. We present this problem, along with several related problems, and develop heuristic algorithms that find near optimal solutions for the bounded cycle-cover problem based on solution techniques for these related problems. Empirical results of these algorithms, applied to randomly generated problem instances, are presented and discussed.

On the average Steiner distance of graphs with prescribed properties

The average n-distance of a connected graph G, μ n ( G), is the average of the Steiner distances of all n-sets of vertices of G. In this paper, we give bounds on μ n for two-connected graphs and for k-chromatic graphs. Moreover, we show that μ n ( G) does not depend on the n-diameter of G.

The Structure of Locally Finite Two-Connected Graphs

We expand on Tutte's theory of $3$-blocks for $2$-connected graphs, generalizing it to apply to infinite, locally finite graphs, and giving necessary and sufficient conditions for a labeled tree to be the $3$-block tree of a $2$-connected graph.

Small Circuit Double Covers of Cubic Multigraphs

Let G be a two-connected graph. A family F of circuits of G is called a circuit double cover (CDC) if each edge of G is contained in exactly two circuits of F. In this paper, we show that if a simple cubic graph G (G ≠ K4) of order n has a CDC, then G has a CDC containing at most n/2 circuits. This result establishes the equivalence of the circuit double cover conjecture (due to Szekeres, Seymour) and the small circuit double cover conjecture (due to Bondy) for any cubic graph. Actually, a stronger result is obtained in this paper for all loopless cubic graphs. Another result in this paper establishes an upper bound on the size of any CDC of a cubic graph.

Minimum-weight two-connected spanning networks

We consider the problem of constructing a minimum-weight, two-connected network spanning all the points in a setV. We assume a symmetric, nonnegative distance functiond(·) defined onV × V which satisfies the triangle inequality. We obtain a structural characterization of optimal solutions. Specifically, there exists an optimal two-connected solution whose vertices all have degree 2 or 3, and such that the removal of any edge or pair of edges leaves a bridge in the resulting connected components. These are the strongest possible conditions on the structure of an optimal solution since we also show thatany two-connected graph satisfying these conditions is theunique optimal solution for a particular choice of ‘canonical’ distances satisfying the triangle inequality. We use these properties to show that the weight of an optimal traveling salesman cycle is at most 4/3 times the weight of an optimal two-connected solution; examples are provided which approach this bound arbitrarily closely. In addition, we obtain similar results for the variation of this problem where the network need only span a prespecified subset of the points.

Counting unlabelled three-connected and homeomorphically irreducible two-connected graphs

Recursive procedures are obtained for counting isomorphism classes of three-connected graphs and of two-connected graphs without vertices of degree 2. We apply an enumeration tool developed by R. W. Robinson to count non-isomorphic 2-connected graphs: he expressed the sums of cycle indices of automorphism groups of connected graphs in terms of those of their 2-connected components, and we do the same for 2-connected graphs and their 3-connected components.

Counting labelled three-connected and homeomorphically irreducible two-connected graphs

Labelled three-connected graphs and labelled two-connected graphs with no vertices of degree 2 are counted using methods similar to those used by Riddell to count labelled two-connected graphs.

Recursive Graphs, Recursive Labelings and Shortest Paths

We consider classes of undirected, not weighted graphs which have recursive representations; these include trees, maximal outplanar graphs, k-trees, chordal graphs, and minimally two-connected graphs. We investigate invariants of recursive labelings of some of these graphs. One consequence of the existence of such invariant relation is that we can describe a single-source, shortest-paths spanning tree in terms of the recursive representation. We also discuss reasons why we cannot do this as well for other types of recursive graphs.

A direct high-temperature star graph expansion for the fourth-field derivative of the Ising free energy

It is shown that the fourth-field derivative, chi 0(2), of the free energy of the Ising model above the critical temperature can be expressed as an expansion in terms of two-connected graphs or stars. The contribution of any star graph to the expansion is expressed in terms of the weak lattice constant of the graph, and a weight, the latter being a property of the star graph only. The method is used to derive series expansions for chi 0(2) on the four-dimensional hyper-simple cubic, hyper-body-centred cubic and hyper-face-centred cubic lattices to order 16, 11 and 9 respectively.

Hamiltonian cycles in regular graphs of moderate degree

In this paper we prove that if k is an integer no less than 3, and if G is a two-connected graph with 2n a vertices, a E {0, 1}, which is regular of degree n k, then G is Hamiltonian if a = 0 and n > k2 + k + 1 or if a = I and n > 2k 2 3k =, 3 . We use the notation and terminology of [1] . Gordon [4] has proved that there are only a small number of exceptional graphs with 2n vertices which are not Hamiltonian when all vertices have degree n 1 or more. The present authors proved [3] that if G is a two-connected graph with 2n vertices which is regular of degree n 2 and if n > 6, then G is Hamiltonian. We now partially extend that result to regular graphs of degree n k, k > 3 . Throughout this paper we suppose that G is a graph with 2n a vertices, with a c {0, 1), which is two connected and regular of degree n k, where k is an integer no less than three . Let P be a longest cycle in G, choose a direction around P, let R = V(G) V(P), and let r = I R 1 . For the lemmas, suppose r > 1 . By a theorem of Dirac [2], ((P) , 2n 2k. For v c R, let C„ be the set of vertices of P adjacent to v, let A,; be the set of vertices of P immediately preceding elements of C, in the ordering of P, and let B,, be the set of vertices of P immediately following elements of C,, . The first lemma is trivial .