scholarly journals Note on the Subgraph Component Polynomial

10.37236/3953 ◽  
2014 ◽  
Vol 21 (3) ◽  
Author(s):  
Yunhua Liao ◽  
Yaoping Hou
Keyword(s):  
Independence Number ◽  
Tutte Polynomial ◽  
Connected Components ◽  
Graph Invariants ◽  
Induced Subgraphs ◽  
Open Problems ◽  
Simple Graphs ◽  
Underlying Graph ◽  
Number Of Cycles ◽  
Regular Bipartite Graph

Tittmann, Averbouch and Makowsky [The enumeration of vertex induced subgraphs with respect to the number of components, European J. Combin. 32 (2011) 954-974] introduced the subgraph component polynomial $Q(G;x,y)$ of a graph $G$, which counts the number of connected components in vertex induced subgraphs. This polynomial encodes a large amount of combinatorial information about the underlying graph, such as the order, the size, and the independence number. We show that several other graph invariants, such as the connectivity and the number of cycles of length four in a regular bipartite graph are also determined by the subgraph component polynomial. Then, we prove that several well-known families of graphs are determined by the polynomial $Q(G;x,y).$ Moreover, we study the distinguishing power and find simple graphs which are not distinguished by the subgraph component polynomial but distinguished by the characteristic polynomial, the matching polynomial and the Tutte polynomial. These are partial answers to three open problems proposed by Tittmann et al.

scholarly journals A complete classification of the complexity of the vertex 3-colourability problem for quadruples of induced 5-vertex prohibitions

2020 ◽  
Vol 22 (1) ◽  
pp. 38-47
Author(s):  
Dmitry S. Malyshev
Keyword(s):  
Graph Theory ◽  
Computational Complexity ◽  
Complete Classification ◽  
Induced Subgraphs ◽  
Simple Graphs ◽  
Forbidden Induced Subgraphs ◽  
The Third ◽  
Hereditary Class ◽  
Complexity Status

The vertex 3-colourability problem for a given graph is to check whether it is possible to split the set of its vertices into three subsets of pairwise non-adjacent vertices or not. A hereditary class of graphs is a set of simple graphs closed under isomorphism and deletion of vertices; the set of its forbidden induced subgraphs defines every such a class. For all but three the quadruples of 5-vertex forbidden induced subgraphs, we know the complexity status of the vertex 3-colourability problem. Additionally, two of these three cases are polynomially equivalent; they also polynomially reduce to the third one. In this paper, we prove that the computational complexity of the considered problem in all of the three mentioned classes is polynomial. This result contributes to the algorithmic graph theory.

scholarly journals Matching Number, Independence Number, and Covering Vertex Number of Γ(Zn)

Mathematics ◽  
2019 ◽  
Vol 7 (1) ◽  
pp. 49
Author(s):  
Eman AbuHijleh ◽  
Mohammad Abudayah ◽  
Omar Alomari ◽  
Hasan Al-Ezeh
Keyword(s):  
Zero Divisor ◽  
Independence Number ◽  
Independent Set ◽  
Covering Problem ◽  
Matching Number ◽  
Graph Invariants ◽  
Vertex Number ◽  
Zero Divisor Graph ◽  
Vertex Covering ◽  
Number Problem

Graph invariants are the properties of graphs that do not change under graph isomorphisms, the independent set decision problem, vertex covering problem, and matching number problem are known to be NP-Hard, and hence it is not believed that there are efficient algorithms for solving them. In this paper, the graph invariants matching number, vertex covering number, and independence number for the zero-divisor graph over the rings Z p k and Z p k q r are determined in terms of the sets S p i and S p i q j respectively. Accordingly, a formula in terms of p , q , k , and r, with n = p k , n = p k q r is provided.

scholarly journals Arithmetical structures on graphs with connectivity one

2018 ◽  
Vol 17 (08) ◽  
pp. 1850147 ◽  
Author(s):  
Hugo Corrales ◽  
Carlos E. Valencia
Keyword(s):  
Positive Integer ◽  
Adjacency Matrix ◽  
Connected Components ◽  
Induced Subgraphs ◽  
Cut Vertex ◽  
One To One

Given a graph [Formula: see text], an arithmetical structure on [Formula: see text] is a pair of positive integer vectors [Formula: see text] such that [Formula: see text] and [Formula: see text] where [Formula: see text] is the adjacency matrix of [Formula: see text]. We describe the arithmetical structures on graph [Formula: see text] with a cut vertex [Formula: see text] in terms of the arithmetical structures on their blocks. More precisely, if [Formula: see text] are the induced subgraphs of [Formula: see text] obtained from each of the connected components of [Formula: see text] by adding the vertex [Formula: see text] and their incident edges, then the arithmetical structures on [Formula: see text] are in one to one correspondence with the [Formula: see text]-rational arithmetical structures on the [Formula: see text]’s. A rational arithmetical structure corresponds to an arithmetical structure where some of the integrality conditions are relaxed.

scholarly journals Forbidden Subgraphs of Power Graphs

10.37236/9961 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Pallabi Manna ◽  
Peter J. Cameron ◽  
Ranjit Mehatari
Keyword(s):  
Nilpotent Groups ◽  
Chordal Graphs ◽  
Forbidden Subgraphs ◽  
Induced Subgraphs ◽  
Open Problems ◽  
Split Graph ◽  
Forbidden Induced Subgraphs ◽  
Power Graph ◽  
Graph Classes ◽  
Split Graphs

The undirected power graph (or simply power graph) of a group $G$, denoted by $P(G)$, is a graph whose vertices are the elements of the group $G$, in which two vertices $u$ and $v$ are connected by an edge between if and only if either $u=v^i$ or $v=u^j$ for some $i$, $j$. A number of important graph classes, including perfect graphs, cographs, chordal graphs, split graphs, and threshold graphs, can be defined either structurally or in terms of forbidden induced subgraphs. We examine each of these five classes and attempt to determine for which groups $G$ the power graph $P(G)$ lies in the class under consideration. We give complete results in the case of nilpotent groups, and partial results in greater generality. In particular, the power graph is always perfect; and we determine completely the groups whose power graph is a threshold or split graph (the answer is the same for both classes). We give a number of open problems.

scholarly journals Spanning Trees with Many Leaves and Average Distance

10.37236/757 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Ermelinda DeLaViña ◽  
Bill Waller
Keyword(s):  
Lower Bounds ◽  
Spanning Tree ◽  
Spanning Trees ◽  
Average Distance ◽  
Independence Number ◽  
Local Independence ◽  
Open Problems ◽  
Maximum Order ◽  
Bipartite Subgraph ◽  
Number Of Leaves

In this paper we prove several new lower bounds on the maximum number of leaves of a spanning tree of a graph related to its order, independence number, local independence number, and the maximum order of a bipartite subgraph. These new lower bounds were conjectured by the program Graffiti.pc, a variant of the program Graffiti. We use two of these results to give two partial resolutions of conjecture 747 of Graffiti (circa 1992), which states that the average distance of a graph is not more than half the maximum order of an induced bipartite subgraph. If correct, this conjecture would generalize conjecture number 2 of Graffiti, which states that the average distance is not more than the independence number. Conjecture number 2 was first proved by F. Chung. In particular, we show that the average distance is less than half the maximum order of a bipartite subgraph, plus one-half; we also show that if the local independence number is at least five, then the average distance is less than half the maximum order of a bipartite subgraph. In conclusion, we give some open problems related to average distance or the maximum number of leaves of a spanning tree.

CHARACTERIZATION OF CHAOS SCENARIOS WITH PERIODIC INCLUSIONS FOR ONE CLASS OF PIECEWISE-SMOOTH DYNAMICAL MAPS

2011 ◽  
Vol 21 (09) ◽  
pp. 2427-2466 ◽  
Author(s):  
JOHN ALEXANDER TABORDA ◽  
FABIOLA ANGULO ◽  
GERARD OLIVAR
Keyword(s):  
Pulse Width Modulation ◽  
Point Of View ◽  
Piecewise Smooth ◽  
Connected Components ◽  
Open Problems ◽  
Smooth Maps ◽  
Digital Pulse ◽  
Piecewise Smooth Maps ◽  
Strongly Connected ◽  
Nonsmooth Dynamical Systems

In this paper, we study bifurcation scenarios characterized by period-adding cascades with alternating chaos in one class of piecewise-smooth maps (PWS). In this class, the state space is separated in three smooth zones defined by a saturation function. Some power converters controlled by Digital Pulse-Width Modulation (PWM) are physical applications of this class of PWS systems denoted by PWS3. Chaos has virtually been detected and studied in all disciplines, however the characterization problem of chaos scenarios has many open problems, mainly in nonsmooth dynamical systems. Novel bifurcation scenarios have recently been reported such as bandcount adding and bandcount increment scenarios based on the numerical detection of bands (where bands are considered as strongly connected components). However, this approach known as Bandcounter cannot be applied to detect bifurcations in chaos scenarios without crisis bifurcations or to identify topological changes inside of one-band chaos. We have proposed a novel framework named Dynamic Linkcounter approach to characterize chaos and torus breakdown scenarios in PWS systems. In this paper, we report overlapping period-adding cascades interspersed with a dynamic linkcount adding cascade. Each complex dynamic link (CDL) structure is a fingered strange attractor increasing in an arithmetic progression the number of CDL or fingers when a bifurcation parameter is varied. Alternative point of view based on tent-map-like structures is given to illustrate the formation of fingered strange attractors.

On the sum of the generalized distance eigenvalues of graphs

2020 ◽  
pp. 2050100
Author(s):  
Hilal A. Ganie ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Distance Matrix ◽  
Wiener Index ◽  
Index Formula ◽  
Graph Invariants ◽  
Matrix Formula ◽  
Simple Connected Graph ◽  
Eigenvalues Of Graphs ◽  
Generalized Distance

In this paper, we study the generalized distance matrix [Formula: see text] assigned to simple connected graph [Formula: see text], which is the convex combinations of Tr[Formula: see text] and [Formula: see text] and defined as [Formula: see text] where [Formula: see text] and Tr[Formula: see text] denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph [Formula: see text], respectively. Denote with [Formula: see text], the generalized distance eigenvalues of [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text] be, respectively, the sum of [Formula: see text]-largest generalized distance eigenvalues and the sum of [Formula: see text]-smallest generalized distance eigenvalues of [Formula: see text]. We obtain bounds for [Formula: see text] and [Formula: see text] in terms of the order [Formula: see text], the Wiener index [Formula: see text] and parameter [Formula: see text]. For a graph [Formula: see text] of diameter 2, we establish a relationship between the [Formula: see text] and the sum of [Formula: see text]-largest generalized adjacency eigenvalues of the complement [Formula: see text]. We characterize the connected bipartite graph and the connected graphs with given independence number that attains the minimum value for [Formula: see text]. We also obtain some bounds for the graph invariants [Formula: see text] and [Formula: see text].

scholarly journals The Differential on Graph Operator Q(G)

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 751
Author(s):  
Ludwin Basilio ◽  
Jair Simon ◽  
Jesús Leaños ◽  
Omar Cayetano
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Vertex Cover ◽  
Domination Number ◽  
Graph Invariants ◽  
Maximum Value ◽  
Vertex Set ◽  
Simple Connected Graph

If G = ( V ( G ) , E ( G ) ) is a simple connected graph with the vertex set V ( G ) and the edge set E ( G ) , S is a subset of V ( G ) , and let B ( S ) be the set of neighbors of S in V ( G ) ∖ S . Then, the differential of S ∂ ( S ) is defined as | B ( S ) | − | S | . The differential of G, denoted by ∂ ( G ) , is the maximum value of ∂ ( S ) for all subsets S ⊆ V ( G ) . The graph operator Q ( G ) is defined as the graph that results by subdividing every edge of G once and joining pairs of these new vertices iff their corresponding edges are incident in G. In this paper, we study the relations between ∂ ( G ) and ∂ ( Q ( G ) ) . Besides, we exhibit some results relating the differential ∂ ( G ) and well-known graph invariants, such as the domination number, the independence number, and the vertex-cover number.

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