Terminal Vertices Research Articles

Ring Closure Matrix

We consider generalization of the terminal matrix of Zaretsky, in which only distances are listed between terminal vertices of acyclic graphs, to a condensed matrix for cyclic and polycyclic systems. Some 50 years ago Zaretsky has proved that the distance matrix between terminal vertices of acyclic graphs allows full reconstruction of the graph. We consider the question: Is there a subset of vertices in cyclic and polycyclic graphs which would allow from information on their distances a full reconstruction of the graph? As we report in this publication a distance matrix confined to a distance between the set of ring closure vertices may offer a positive answer to the considered problem.

Characteristic Polynomial Followed by Trigonometric Identity for Obtaining Analytical Eigenspectra of Some Weighted Graphs of Linear Chains and Cycles

Abstract Graphs of linear chains and/or cycles with homogeneous and/or alternant vertex weights have been considered. If one puts weight(s) on the terminal vertex (vertices) or edge(s) of such linear chains, another set of linear chains might result. All such vertex- or edge-weighted graphs of linear chains and cycles represent conjugated systems with heteroatom(s). Characteristic polynomial followed by trigonometric identity has been shown to be used for obtaining eigenspectra for such systems of linear chains and cycles in closed analytical forms. The analytical expressions so obtained for such systems can be used for deriving the eigenspectra of some complicated π-conjugated molecules in analytical forms.

Preserving Terminal Distances Using Minors

We introduce the following notion of compressing an undirected graph $G$ with (nonnegative) edge-lengths and terminal vertices $R\subseteq V(G)$. A distance-preserving minor is a minor $G'$ (of $G$) with possibly different edge-lengths, such that $R\subseteq V(G')$ and the shortest-path distance between every pair of terminals is exactly the same in $G$ and in $G'$. We ask: what is the smallest $f^*(k)$ such that every graph $G$ with $k=|R|$ terminals admits a distance-preserving minor $G'$ with at most $f^*(k)$ vertices? Simple analysis shows that $f^*(k)\le O(k^4)$. Our main result proves that $f^*(k)\ge \Omega(k^2)$, significantly improving on the trivial $f^*(k)\ge k$. Our lower bound holds even for planar graphs $G$, in contrast to graphs $G$ of constant treewidth, for which we prove that $O(k)$ vertices suffice.

Tree topology inference from leaf-to-leaf path lengths using Prüfer sequence

In this paper, we propose an algorithm to reconstruct the set of nodes and their topology relationships, particularly in case of a topological tree where all edge weights are 1, given the set of terminal vertices or leaves and their pairwise distances or path lengths. Prufer encoding and decoding techniques showed a one-to-one correspondence between n-labelled trees and n − 2-tuples of vertex labels. We consider Prufer encoding technique in discovering the intermediate vertices from the end-to-end path length between all pairs of terminal vertices. The results are useful in inferring graph characteristics, topology in particular, in the field of network tomography where the internal structure and link-level performance are inferred from end-to-end measurements.

KmsGC: An Unsupervised Color Image Segmentation Algorithm Based onK-Means Clustering and Graph Cut

For unsupervised color image segmentation, we propose a two-stage algorithm, KmsGC, that combinesK-means clustering with graph cut. In the first stage,K-means clustering algorithm is applied to make an initial clustering, and the optimal number of clusters is automatically determined by a compactness criterion that is established to find clustering with maximum intercluster distance and minimum intracluster variance. In the second stage, a multiple terminal vertices weighted graph is constructed based on an energy function, and the image is segmented according to a minimum cost multiway cut. A large number of performance evaluations are carried out, and the experimental results indicate the proposed approach is effective compared to other existing image segmentation algorithms on the Berkeley image database.

On Second Atom-Bond Connectivity Index

The atom-bond connectivity index of graph is a topological index proposed by Estrada et al. as ABC (G)  uvE (G ) (du dv  2) / dudv , where the summation goes over all edges of G, du and dv are the degrees of the terminal vertices u and v of edge uv. In the present paper, some upper bounds for the second type of atom-bond connectivity index are computed.

Moplex orderings generated by the LexDFS algorithm

Let G be a graph with vertex set V. A moplex of G is both a clique and a module whose neighborhood is a minimal separator in G or empty. A moplex ordering of G is an ordered partition (X1,X2,⋯,Xk) of V for some integer k into moplexes which are defined in the successive transitory elimination graphs, i.e., for 1⩽i⩽k−1, Xi is a moplex of the graph Gi induced by ∪j=ikXj and Xk induces a clique. In this paper we prove the terminal vertex by an execution of the lexicographical depth-first search (LexDFS for short) algorithm on G belongs to a moplex whose vertices are numbered consecutively and further that the LexDFS algorithm on G defines a moplex ordering of G, which is similar to the result about the maximum cardinality search (MCS for short) algorithm on chordal graphs [J.R.S. Blair, B.W. Peyton, An introduction to chordal graphs and clique trees, IMA Volumes in Mathematics and its Applications, 56 (1993) pp. 1–30] and the result about the lexicographical breadth-first search (LexBFS for short) algorithm on general graphs [A. Berry, J.-P. Bordat, Separability generalizes Dirac’s theorem, Discrete Appl. Math., 84 (1998) 43–53]. As a corollary, we can obtain a simple algorithm on a chordal graph to generate all minimal separators and all maximal cliques.

On the inverse of the adjacency matrix of a graph

A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two–vertex deleted subgraphs of G and of the graph ⌈(G −1 ) associated with G −1 to remain NSSDs is that the submatrices belonging to them, derived from G and G −1 , are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.

The proof of a conjecture on the lewin number of primitive non-powerful signed digraphs

Suppose S is a primitive non-powerful signed digraph. A pair of SSSD walks of S are two directed walks of S, which have the same initial vertex, same terminal vertex and same length, but different signs. The lewin number of S, denoted l(S), is the least positive integer k such that there are both SSSD walks of lengths k and k+1 from some vertex u to some vertex v (possibly u again) of S. This paper presents a proof of a conjecture on l(S), which was put forward by You et al. [L.H. You, M.H. Liu, B.L. Liu, Bounds on the lewin number for primitive non-powerful signed digraphs, Acta Math. Appl. Sinica 35 (2012) 396–407].

ON THE MAXIMAL DISTANCE SPECTRAL RADIUS IN A CLASS OF BICYCLIC GRAPHS

Bicyclic graphs are connected graphs in which the number of edges equals the number of vertices plus one. Let Pp+1 = x1x2⋯xp+1, Pt+1 = y1y2⋯yt+1 and Pq+1 = z1z2⋯zq+1 be three vertex-disjoint paths. Identifying the initial vertices as u0 and the terminal vertices as v0, the resultant graph, denoted by θ(p; t; q), is called a θ-graph. Let [Formula: see text] be the class of all bicyclic graphs on n vertices, which contain a θ-graph as an induced subgraph. In this paper, we study the distance spectral radius of bicyclic graphs in [Formula: see text], and determine the graph with the largest distance spectral radius.

Primal-dual approximation algorithms for Node-Weighted Steiner Forest on planar graphs

Node-Weighted Steiner Forest is the following problem: Given an undirected graph, a set of pairs of terminal vertices, a weight function on the vertices, find a minimum weight set of vertices that includes and connects each pair of terminals. We consider the restriction to planar graphs where the problem remains NP-complete. Demaine et al. showed that the generic primal-dual algorithm of Goemans and Williamson is a 6-approximation on planar graphs. We present (1) two different analyses to prove an approximation factor of 3, (2) show that our analysis is best possible for the chosen proof strategy, and (3) generalize this result to feedback problems on planar graphs.We give a simple proof for the first result using contraction techniques and following a standard proof strategy for the generic primal-dual algorithm. Given this proof strategy our analysis is best possible which implies that proving a better upper bound for this algorithm, if possible, would require different proof methods. Then, we give a reduction on planar graphs of Feedback Vertex Set to Node-Weighted Steiner Tree, and Subset Feedback Vertex Set to Node-Weighted Steiner Forest. This generalizes our result to the feedback problems studied by Goemans and Williamson. For the opposite direction, we show how our constructions can be combined with the proof idea for the feedback problems to yield an alternative proof of the same approximation guarantee for Node-Weighted Steiner Forest.

The uncapacitated swapping problem on a line and on a circle

The uncapacitated swapping problem is defined by a graph consisting of n vertices, and m object types. Each vertex of the graph is associated with two object types: the one that it currently holds, and the one it demands. Each vertex holds or demands at most one unit of an object. The problem is balanced in the sense that for each object type, its total supply equals its total demand. A vehicle of unlimited capacity is assumed to transport the objects in order to fulfill the requirements of all vertices. The objective is to find a shortest route along which the vehicle can accomplish the rearrangement of the objects, given designated initial and terminal vertices. The uncapacitated swapping problem on a general graph, including a tree graph, is known to be NP-Hard. In this paper we show that for the line and circle graphs, the problem is polynomially solvable: we propose an O(n)-time algorithm for a line and an O(n3)-time algorithm for a circle.

Approach to building inspection tests of digital devices on extra large integrated circuits

Considered the problem of reducing the running in constructing a complete test for digital devices designed with using the VLSI circuits. The process of constructing a test for a given fault is reduced to finding the terminal vertices in the signals assignment tree. To reduce the iteration is proposed the focused search method wich designed to apply as the general theoretical method for finding solutions in productional type systems. Describes the results of experiments with the combinational circuits.

Extremal digraphs whose walks with the same initial and terminal vertices have distinct lengths

Let D be a digraph of order n in which any two walks with the same initial vertex and the same terminal vertex have distinct lengths. We prove that D has at most (n+1)2/4 arcs if n is odd and n(n+2)/4 arcs if n is even. The digraphs attaining this maximum size are determined.

The subgroup graph of a group

Given any subgroup H of a group G, let Γ H (G) be the directed graph with vertex set G such that x is the initial vertex and y is the terminal vertex of an edge if and only if x ≠ y and \({xy\in H}\) . Furthermore, if \({xy\in H}\) and \({yx\in H}\) for some \({x,y\in G}\) with x ≠ y, then x and y will be regarded as being connected by a single undirected edge. In this paper, the structure of the connected components of Γ H (G) is investigated. All possible components are provided in the cases when |H| is either two or three, and the graph Γ H (G) is completely classified in the case when H is a normal subgroup of G and G/H is a finite abelian group.

Fast Multilevel & Always Cancelation-free Method for Large Electric Networks Exact Symbolic Analysis

The paper presents an algorithm of the exact symbolic network function analysis that deals with circuits with any size. The only condition is to decompose the whole circuit into smaller subcircuits. The decomposition is the multi-level hierarchial one. What is more, the calculation for each level can be done only once. One can reuse the partial results any time when the previously determined subcircuit appears in any analyzed structure. A higher level subcircuit needs only a few information about internal structure of a lower level, so the method can be easily implemented in multiprocessor or distributed systems. Although multilevel and compressed structure, symbolical values remain cancelation-free and any path from the root to the terminal vertex 1 represents a single term. Thus, a large-scale and small-scale sensitivities calculation and elimination of less significant terms could be simply and natural.

Fabrication of ZnO Nanorods in One Pot via Solvothermal Method

AbstractIn this article, we reported a low temperature solvothermal method to synthesize ZnO nanorods in one pot. In order to study the growth mechanism of ZnO nanorods, various preparative parameters, such as zinc species concentration, the molar concentration ratio of hydroxide [OH‐] to zinc species [Zn2+], and surfactants, have been systematically examined. According to the analysis, the aspect ratio of ZnO nanorods was enlarged with the increase of the concentration ratio of [OH‐] to [Zn2+]. However, the mean diameter only causes subtle change. Additionally, the influence of surfactants was studied by two different kinds of surfactant, ethylenediamine (EDA) and hexadecylamine (HDA). From transmission electron microscopy analysis, the aspect ratio of ZnO nanorods was similar to that without adding surfactant. However, the green emission caused by oxygen defects or others can be omitted while the surfactants were added into the reaction. Herein, the growth mechanism was proposed to relate to the element of coordination polyhedron presented in the interface. When terminal vertex bonds with more growth units, the interface has higher growth rate for the crystal formation. Besides, the reaction medium also affects the growth mechanism of ZnO nanorods. In an alkali medium, the larger hindrance effect of ONa‐ ions at the interface of (0001) and ( 01 1¯ 1¯ ) faces, the growth rates of (0001) and ( 01 1¯ 1¯ ) faces have a great hindrance relative to other crystal faces.

Nash-solvable two-person symmetric cycle game forms

A two-person positional game form g (with perfect information and without moves of chance) is modeled by a finite directed graph (digraph) whose vertices and arcs are interpreted as positions and moves, respectively. All simple directed cycles of this digraph together with its terminal positions form the set A of the outcomes. Each non-terminal position j is controlled by one of two players i ∈ I = { 1 , 2 } . A strategy x i of a player i ∈ I involves selecting a move ( j , j ′ ) in each position j controlled by i . We restrict both players to their pure positional strategies; in other words, a move ( j , j ′ ) in a position j is deterministic (not random) and it can depend only on j (not on preceding positions or moves or on their numbers). For every pair of strategies ( x 1 , x 2 ) , the selected moves uniquely define a play, that is, a directed path form a given initial position j 0 to an outcome (a directed cycle or terminal vertex). This outcome a ∈ A is the result of the game corresponding to the chosen strategies, a = a ( x 1 , x 2 ) . Furthermore, each player i ∈ I = { 1 , 2 } has a real-valued utility function u i over A . Standardly, a game form g is called Nash-solvable if for every u = ( u 1 , u 2 ) the obtained game ( g , u ) has a Nash equilibrium (in pure positional strategies). A digraph (and the corresponding game form) is called symmetric if ( j , j ′ ) is its arc whenever ( j ′ , j ) is. In this paper we obtain necessary and sufficient conditions for Nash-solvability of symmetric cycle two-person game forms and show that these conditions can be verified in linear time in the size of the digraph.

A global Riesz decomposition theorem on trees without positive potentials

The functions H and Hv, which first appear in [3, pp. 5–6], play a major role throughout this paper. One of the stated properties of Hv is that it is harmonic everywhere except at v. However, Hv fails to be harmonic at any terminal vertex. Accordingly, we assume that the tree has no terminal vertices. With this assumption, all stated properties and results of [3] are correct. This error was first observed in connection with [1] (cf. [2]) where the functions Hv were first introduced.

Clique or hole in claw-free graphs

Given a claw-free graph and two non-adjacent vertices x and y without common neighbours we prove that there exists a hole through x and y unless the graph contains the obvious obstruction, namely a clique separating x and y. We derive two applications: We give a necessary and sufficient condition for the existence of an induced x– z path through y, where x , y , z are prescribed vertices in a claw-free graph; and we prove an induced version of Mengerʼs theorem between four terminal vertices. Finally, we improve the running time for detecting a hole through x and y and for the Three-in-a-Tree problem, if the input graph is claw-free.