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Chromatic polynomials of generalized trees

Earl Glen Whitehead
1988 Discrete Mathematics  
This paper is a survey of results on chromatic polynomials of graphs which are generalizations of trees. In particular, chromatic polynomials of q-trees will be discussed.  ...  Another generalization of trees is the n-gon-trees. The smallest n-gon-tree (n * 3) is the n-gon which is a cycle on n vertices.  ...  no subgraphs homeomorphic to &, and a careful analysis of the coefficients of the chromatic polynomial of a n-gon-tree.  ... 
doi:10.1016/0012-365x(88)90231-2 fatcat:v5opf5ppevetjekqjjs2nkeayi

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs [article]

Boris Brimkov, Illya V. Hicks
2015 arXiv   pre-print
We present a low-order polynomial time algorithm for finding the chromatic polynomials of generalized vertex joins of trees, and by duality we find the flow polynomials of arbitrary outerplanar graphs.  ...  We also present closed formulas for the chromatic and flow polynomials of vertex joins of cliques and cycles, otherwise known as "generalized wheel" graphs.  ...  For instance, in the next section, we compute the chromatic polynomials of generalized vertex joins of trees.  ... 
arXiv:1501.04388v1 fatcat:owu5rz5vlbasrooos7kqz2nyf4

A new two-variable generalization of the chromatic polynomial

Klaus Dohmen, André Poenitz, Peter Tittmann
2003 Discrete Mathematics & Theoretical Computer Science  
We finally give explicit expressions for the generalized chromatic polynomial of complete graphs, complete bipartite graphs, paths, and cycles, and show that it can be computed in polynomial time for trees  ...  International audience We present a two-variable polynomial, which simultaneously generalizes the chromatic polynomial, the independence polynomial, and the matching polynomial of a graph.  ...  In particular, the derivations of the generalized chromatic polynomial of a path (Subsection 5.3) and a cycle (Subsection 5.4) were greatly simplified by one of the referees, whose solution we adopted  ... 
doi:10.46298/dmtcs.335 fatcat:v4p2i5k7fbhajlhlazksxj5j2i

Chromatic Polynomial of Domination Subdivision Non Stable Graphs

2019 VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE  
In this paper, we provide a method of determining the chromatic polynomial of DSNS graph from G.  ...  Yamuna et al. have determined the domination number of G  , G  , chromatic polynomial of G  , spanning tree of G  , number of spanning trees of G  from G. 5 . 5 The graph G in Fig.1is DSNS.  ...  INTRODUCTION In [1] , Shubo Chen has investigated absolute sum of chromatic polynomial coefficients of forest, q -tree, unicyclic graphs and quasi wheel graphs.  ... 
doi:10.35940/ijitee.k2402.1081219 fatcat:biidduynfrhd7br6r53zlj3iuu

Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

Boris Brimkov, Illya V. Hicks
2016 Discrete Applied Mathematics  
We present a low-order polynomial time algorithm for computing the chromatic polynomials of generalized vertex joins of trees; by duality, this algorithm can also be used to compute the flow polynomials  ...  We also present closed formulas for the chromatic polynomials of generalized vertex joins of cliques, and the chromatic and flow polynomials of generalized vertex joins of cycles.  ...  For instance, in the next section, we compute the chromatic polynomials of generalized vertex joins of trees.  ... 
doi:10.1016/j.dam.2015.10.016 fatcat:ceyvgshtw5g7ncj75j7ddkw5ay

The computation of chromatic polynomials

Gary Haggard, Thomas R. Mathies
1999 Discrete Mathematics  
The computation of the chromatic polynomial of this graph is computed by enhancing the algorithm based on the classical Delete-Contract theorem as well as finding approaches for substantially modifying  ...  a computation tree during computation.  ...  Fig. 1 . 1 The huncated 2 . 2 The chromatic polynomial The chromatic polynomial of TI is given in terms of the tree basis, {,%(A -1)' 1 i= 1,2,..., 58,59}.  ... 
doi:10.1016/s0012-365x(98)00343-4 fatcat:2txofzddj5fjtggrygtrydgdau

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

Pascal Berthomé, Sylvain Lebresne, Kim Nguyễn
2005 Lecture Notes in Computer Science  
In this paper, we present a new algorithm for computing the chromatic polynomial of a general graph G.  ...  Furthermore, we give some lower bounds of the general complexity of our method, and provide experimental results for several families of graphs.  ...  Acknowledgments The authors thank Ioan Todinca for the counterexample of Figure 7 .  ... 
doi:10.1007/11604686_32 fatcat:s6ea76zshjcq3a6txx4bivs2gq

Extension of Chromatic Polynomials by Utilizing Mobius Inversion Theorem

R.V.N. SrinivasaRao, J. VenkateswaraRao, Haftamu Menker GebreYohannes
2013 Contemporary Mathematics and Statistics  
This manuscript determines the chromatic polynomial for various graphs using Mobius inversion theorem, similar to the combinatorial proofs.  ...  It also applies contraction-deletion algorithm to derive the polynomial identities and show that these polynomial identities are matching with the identities obtained from Mobius inversion theorem.  ...  In general, if we want the tree of order k, 4 kn  we can contract in ways.  ... 
doi:10.7726/cms.2013.1005 fatcat:b3w4idbmindxdncaxshkip5ewq

Author index to volume 172 (1997)

1997 Discrete Mathematics  
Chromatic uniqueness of the complements of certain forests Li, N.-Z., The list of chromatically unique graphs of order seven and eight  ...  Wang, Chromatic equivalence classes of certain generalized polygon trees Rodriguez, J. and A. Satyanarayana, Chromatic polynomials with least coefficients Sakaloglu, A. and A.  ...  ., Adjoint polynomials and chromatically unique graphs Lu, Z., The exact value of the harmonious chromatic number of a complete binary tree Ouyang, K.Z., see X.E. Chen Ouyang, K.Z., see X.E.  ... 
doi:10.1016/s0012-365x(97)84106-4 fatcat:3hvrb2m7ojgrvj6gae4mzysv5m

A Graph Polynomial from Chromatic Symmetric Functions [article]

William Chan, Logan Crew
2022 arXiv   pre-print
basis given by chromatic symmetric functions of trees.  ...  This paper describes how many known graph polynomials arise from the coefficients of chromatic symmetric function expansions in different bases, and studies a new polynomial arising by expanding over a  ...  This research was sponsored by the National Sciences and Engineering Research Council of Canada.  ... 
arXiv:2110.15291v3 fatcat:wqk2csgwkjaghfqqqodtovcy3i

Chromatic polynomaials for regular graphs and modified wheels

Beatrice Loerinc, Earl Glen Whitehead
1981 Journal of combinatorial theory. Series B (Print)  
We show that the coefficients of chromatic polynomials of certain connected graphs, relative to the tree basis, do not exhibit the strong logarithmic concavity property.  ...  Let P(G;2) denote the chromatic polynomial of a graph G, expressed in the variable 2. A graph G is chromatically unique if P(G; 2) = P(H.; 2) implies that H is isomorphic to G.  ...  In [7] , Nijenhuis and Witf call the expansion of a chromatic polynomial P(G; 2) in the tree basis "the Tutte polynomial form" of P(G;2).  ... 
doi:10.1016/s0095-8956(81)80010-x fatcat:bredl3kye5gnpdpqvgqvdoidrq

On the real roots of σ-Polynomials [article]

Jason Brown, Aysel Erey
2016 arXiv   pre-print
These polynomials are closely related to chromatic polynomials, as the chromatic polynomial of G is given by ∑_i=χ(G)^n a_i(G) x(x-1) ... (x-(i-1)).  ...  It is known that the closure of the real roots of chromatic polynomials is precisely {0, 1} [32/27,∞), with (-∞,0), (0,1) and (1,32/27) being maximal zero-free intervals for roots of chromatic polynomials  ...  Obviously, trees are triangle-free graphs. Also, as we already mentioned, the matching polynomials of trees and equal to their characteristic polynomials [8] .  ... 
arXiv:1611.09525v1 fatcat:qp5uyvuykfhnnkzh6g47hdq3zi

Some polynomials of flower graphs

E. Mphako-Banda
2007 International Mathematical Forum  
We define a class of graphs called flower and give some properties of these graphs. Then the explicit expressions of the chromatic polynomial and the flow polynomial is given.  ...  Further, we give classes of graphs with the same chromatic and flow polynomials. Mathematics Subject Classification: 05C99  ...  Finally, we give a general set of non-isomorphic graphs obtained from flower graphs, having the same chromatic and flow polynomials.  ... 
doi:10.12988/imf.2007.07221 fatcat:jabdw4nibnbh3pjx6xvtje3hyq

Page 6 of Mathematical Reviews Vol. , Issue 2004b [page]

2004 Mathematical Reviews  
The author considers simply generated families of trees with no vertices of out-degree zero.  ...  edges of Ks, has chromatic polynomial P(H7, 4) = AA —1)(A —2)(4 — 3)3 (4 — 4), and it is not triangulated.  ... 

Computing Graph Polynomials on Graphs of Bounded Clique-Width [chapter]

J. A. Makowsky, Udi Rotics, Ilya Averbouch, Benny Godlin
2006 Lecture Notes in Computer Science  
We show that the chromatic polynomial, the matching polynomial and the two-variable interlace polynomial of a graph G of clique-width at most k with n vertices can be computed in time O(n f (k) ), where  ...  f (k) ≤ 3 for the inerlace polynomial, f (k) ≤ 2k + 1 for the matching polynomial and f (k) ≤ 3 · 2 k+2 for the chromatic polynomial.  ...  Interesting generalizations of the chromatic polynomial were introduced by H. Whitney in 1932 and Tutte in 1947.  ... 
doi:10.1007/11917496_18 fatcat:wx5yb77xd5g2dgvd3lufnhkdqy
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