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Chordal graphs and the characteristic polynomial

2003
*
Discrete Mathematics
*

In particular,

doi:10.1016/s0012-365x(02)00500-9
fatcat:qkvt73dhvzgtngi4v27sfhqqy4
*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*. ...##
###
Graphs whose flow polynomials have only integral roots
[article]

2009
*
arXiv
*
pre-print

We show if

arXiv:0908.0181v2
fatcat:ctooxct4pfh4zlp4codahacnoq
*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. ...##
###
Graphs whose flow polynomials have only integral roots

2011
*
European journal of combinatorics (Print)
*

We show if

doi:10.1016/j.ejc.2011.01.010
fatcat:pxsdxmaxlfcsvmmjmplozrshwa
*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. ...##
###
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]

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. ...

##
###
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

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*. ...

##
###
Harmonious Coloring on Subclasses of Colinear Graphs
[chapter]

2010
*
Lecture Notes in Computer Science
*

be NP-complete on both split

doi:10.1007/978-3-642-11440-3_13
fatcat:lcuo2w4q5jh75b6noqdj4n4dtu
*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*. ...##
###
Sums of Polynomials and Clique Roots
[article]

2021
*
arXiv
*
pre-print

necessary condition for having a real root for sums of

arXiv:2112.09744v1
fatcat:ws4sszglzzhblm7vq6nic2vfpm
*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 ...##
###
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

1994
*
Discrete Mathematics
*

In this paper, we will introduce

doi:10.1016/0012-365x(94)90393-x
fatcat:wtz3bn44rzcbvp5yy5oc3o3cxq
*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. ...##
###
Supersolvable Frame-matroid and Graphic-lift Lattices

2001
*
European journal of combinatorics (Print)
*

We apply

doi:10.1006/eujc.2000.0418
fatcat:7mjlmlonazafbbnqp4sue2ugtu
*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. ...##
###
Learning restricted Bayesian network structures
[article]

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. ...

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

2005
*
Lecture Notes in Computer Science
*

Our method is based on

doi:10.1007/11604686_32
fatcat:s6ea76zshjcq3a6txx4bivs2gq
*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 . ...
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