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Chordal graphs and the characteristic polynomial

Elizabeth W. McMahon, Beth A. Shimkus, Jessica A. Wolfson
2003 Discrete Mathematics  
In particular, the characteristic polynomial for a chordal graph is an alternating clique generating function and is expressible in terms of the clique decomposition of the graph.  ...  When chordal graphs are viewed as antimatroids by shelling of simplicial vertices, the greedoid characteristic polynomial gives additional information about those graphs.  ...  We begin our investigation of the characteristic polynomial of chordal graphs with the characteristic polynomial of a complete graph.  ... 
doi:10.1016/s0012-365x(02)00500-9 fatcat:qkvt73dhvzgtngi4v27sfhqqy4

Graphs whose flow polynomials have only integral roots [article]

Joseph P.S. Kung, Gordon F. Royle
2009 arXiv   pre-print
We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph.  ...  Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.  ...  The authors gratefully acknowledge the contribution of Maria Monks who alerted us to an error in an earlier version of this paper.  ... 
arXiv:0908.0181v2 fatcat:ctooxct4pfh4zlp4codahacnoq

Graphs whose flow polynomials have only integral roots

Joseph P.S. Kung, Gordon F. Royle
2011 European journal of combinatorics (Print)  
We show if the flow polynomial of a bridgeless graph G has only integral roots, then G is the dual graph to a planar chordal graph.  ...  Expressed in the language of matroid theory, this result says that the cographic matroids with only integral characteristic roots are the cycle matroids of planar chordal graphs.  ...  Acknowledgement The authors gratefully acknowledge the contribution of Maria Monks, who alerted us to an error in an earlier version of this paper.  ... 
doi:10.1016/j.ejc.2011.01.010 fatcat:pxsdxmaxlfcsvmmjmplozrshwa

Page 7306 of Mathematical Reviews Vol. , Issue 2003j [page]

2003 Mathematical Reviews  
, PA); Wolfson, Jessica A. (1-LFYC; Easton, PA) Chordal graphs and the characteristic polynomial.  ...  The authors study this anti- matroid characteristic polynomial for the class of chordal graphs. A graph is chordal if every cycle of length 4 or greater has a chord.  ... 

On Distance-d Independent Set and other problems in graphs with few minimal separators [article]

Pedro Montealegre, Ioan Todinca
2016 arXiv   pre-print
The problem is NP-hard on chordal graphs for any odd d ≥ 3.  ...  We also provide polynomial algorithms for Connected Vertex Cover and Connected Feedback Vertex Set on subclasses of including chordal and circular-arc graphs, and we discuss variants of independent domination  ...  We thank Iyad Kanj for fruitful discussions on the subject.  ... 
arXiv:1607.04545v1 fatcat:s3y6oshzzfhhdbfnufiyv43ide

Page 810 of Mathematical Reviews Vol. , Issue 2002B [page]

2002 Mathematical Reviews  
As chordal graphs can be colored in polynomial time, the coloring problem for even powers of chordal graphs becomes more difficult.  ...  In addition, the authors provide new and constructive proofs of the results that an odd power of a chordal graph is chordal, and any power of an interval graph is an interval graph.  ... 

Finding dominating cliques efficiently, in strongly chordal graphs and undirected path graphs

Dieter Kratsch
1990 Discrete Mathematics  
The minimum dominating clique problem is NP-complete for split graphs and, hence, for chordal graphs.  ...  We show that for two other important subclasses of chordal graphs the problem is solvable efficiently.  ...  DC is solvable in polynomial time for a lot of special classes of perfect graphs, e.g. comparability graphs and chordal graphs.  ... 
doi:10.1016/0012-365x(90)90363-m fatcat:27o5er5wczcirnbo5vmnx4ff64

Harmonious Coloring on Subclasses of Colinear Graphs [chapter]

Kyriaki Ioannidou, Stavros D. Nikolopoulos
2010 Lecture Notes in Computer Science  
be NP-complete on both split and strongly chordal graphs.  ...  Extending previous NP-completeness results of the harmonious coloring problem on subclasses of chordal and co-chordal graphs, we prove that the problem remains NP-complete for split undirected path graphs  ...  The problem is NPcomplete on co-chordal graphs, since it is NP-complete on split graphs [1] , and it has a polynomial solution on threshold graphs.  ... 
doi:10.1007/978-3-642-11440-3_13 fatcat:lcuo2w4q5jh75b6noqdj4n4dtu

Sums of Polynomials and Clique Roots [article]

Hossein Teimoori Faal
2021 arXiv   pre-print
necessary condition for having a real root for sums of polynomials with (at least) one real root.  ...  Finally, we conclude our paper with several interesting open problems and conjectures for interested readers.  ...  For instance, the first derivatives of several graph polynomials like characteristic polynomial, matching polynomial, independence polynomial and the clique polynomials of a given graph can be obtained  ... 
arXiv:2112.09744v1 fatcat:ws4sszglzzhblm7vq6nic2vfpm

Page 4473 of Mathematical Reviews Vol. , Issue 95h [page]

1995 Mathematical Reviews  
The authors compute topological characteristics for all polyhedral graphs up to 8 vertices including the characteristic polynomial, the distance polynomial, the Z-counting polynomial, the spectrum, the  ...  The dominating set problem and the min- imum dominating clique problem are shown to be equivalent for strongly chordal graphs, and both are equivalent to the problem of finding a minimum subset of the  ... 

Page 2132 of Mathematical Reviews Vol. , Issue 97D [page]

1997 Mathematical Reviews  
; Lanzhou) A solution to Gutman’s problem on the characteristic polynomial of a bipartite graph.  ...  Summary: “A graph G is called integral if all the zeros of the characteristic polynomial P(G; A) are integers.  ... 

The generating polynomial and Euler characteristic of intersection graphs

Yeong-Nan Yeh
1994 Discrete Mathematics  
In this paper, we will introduce the generating polynomial E,(x) and the Euler characteristic e(G) of a graph G. We will study several various operations performing on two graphs (surfaces).  ...  An immediate consequence is that the four contractible transformations do not change the Euler characteristic of a graph. Furthermore, we prove that all chordal graphs are contractible.  ...  Acknowledgment The author is indebted to the National Science Council, Taiwan, Republic of China, for its financial support under Grant NSC-82-0208-MOOl-42.  ... 
doi:10.1016/0012-365x(94)90393-x fatcat:wtz3bn44rzcbvp5yy5oc3o3cxq

Supersolvable Frame-matroid and Graphic-lift Lattices

Thomas Zaslavsky
2001 European journal of combinatorics (Print)  
We apply the characterizations to three kinds of example: one generalizes the root system D n and near-Dowling lattices; one has for characteristic polynomials the enumerators of separated circular partial  ...  Using the fact that nitary frame matroids are the bias matroids of biased graphs, we characterize modular coatoms in frames of nite rank and we describe explicitly the frames that are supersolvable.  ...  Acknowledgement I thank Marge Pratt for expert and patient typing of many and messy changes to get the manuscript completed.  ... 
doi:10.1006/eujc.2000.0418 fatcat:7mjlmlonazafbbnqp4sue2ugtu

Learning restricted Bayesian network structures [article]

Raymond Hemmecke, Silvia Lindner, Milan Studený
2010 arXiv   pre-print
graph explains the occurrence of the observed data.  ...  For this, we introduce a new unique algebraic representative for these structures, called the characteristic imset.  ...  Acknowledgements The results of Milan Studený have been supported by the grants GAČR n. 201/08/0539 and GAAVČR n. IAA100750603.  ... 
arXiv:1011.6664v1 fatcat:sfdgoharzrgvjbyqgfmxvsx4wm

Computation of Chromatic Polynomials Using Triangulations and Clique Trees [chapter]

Pascal Berthomé, Sylvain Lebresne, Kim Nguyễn
2005 Lecture Notes in Computer Science  
Our method is based on the addition of edges and contraction of non-edges of G, the base case of the recursion being chordal graphs.  ...  In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G.  ...  Acknowledgments The authors thank Ioan Todinca for the counterexample of Figure 7 .  ... 
doi:10.1007/11604686_32 fatcat:s6ea76zshjcq3a6txx4bivs2gq
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