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Chromatic polynomials of generalized trees

1988
*
Discrete Mathematics
*

This paper is a survey

doi:10.1016/0012-365x(88)90231-2
fatcat:v5opf5ppevetjekqjjs2nkeayi
*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*. ...##
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Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs
[article]

2015
*
arXiv
*
pre-print

We present a low-order

arXiv:1501.04388v1
fatcat:owu5rz5vlbasrooos7kqz2nyf4
*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*. ...##
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A new two-variable generalization of the chromatic polynomial

2003
*
Discrete Mathematics & Theoretical Computer Science
*

We finally give explicit expressions for the

doi:10.46298/dmtcs.335
fatcat:v4p2i5k7fbhajlhlazksxj5j2i
*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 ...##
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Chromatic Polynomial of Domination Subdivision Non Stable Graphs

2019
*
VOLUME-8 ISSUE-10, AUGUST 2019, REGULAR ISSUE
*

In this paper, we provide a method

doi:10.35940/ijitee.k2402.1081219
fatcat:biidduynfrhd7br6r53zlj3iuu
*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. ...##
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Chromatic and flow polynomials of generalized vertex join graphs and outerplanar graphs

2016
*
Discrete Applied Mathematics
*

We present a low-order

doi:10.1016/j.dam.2015.10.016
fatcat:ceyvgshtw5g7ncj75j7ddkw5ay
*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*. ...##
###
The computation of chromatic polynomials

1999
*
Discrete Mathematics
*

The computation

doi:10.1016/s0012-365x(98)00343-4
fatcat:2txofzddj5fjtggrygtrydgdau
*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}. ...##
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Computation of Chromatic Polynomials Using Triangulations and Clique Trees
[chapter]

2005
*
Lecture Notes in Computer Science
*

In this paper, we present a new algorithm for computing the

doi:10.1007/11604686_32
fatcat:s6ea76zshjcq3a6txx4bivs2gq
*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 . ...##
###
Extension of Chromatic Polynomials by Utilizing Mobius Inversion Theorem

2013
*
Contemporary Mathematics and Statistics
*

This manuscript determines the

doi:10.7726/cms.2013.1005
fatcat:b3w4idbmindxdncaxshkip5ewq
*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. ...##
###
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. ...

##
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A Graph Polynomial from Chromatic Symmetric Functions
[article]

2022
*
arXiv
*
pre-print

basis given by

arXiv:2110.15291v3
fatcat:wqk2csgwkjaghfqqqodtovcy3i
*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. ...##
###
Chromatic polynomaials for regular graphs and modified wheels

1981
*
Journal of combinatorial theory. Series B (Print)
*

We show that the coefficients

doi:10.1016/s0095-8956(81)80010-x
fatcat:bredl3kye5gnpdpqvgqvdoidrq
*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). ...##
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On the real roots of σ-Polynomials
[article]

2016
*
arXiv
*
pre-print

These

arXiv:1611.09525v1
fatcat:qp5uyvuykfhnnkzh6g47hdq3zi
*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] . ...##
###
Some polynomials of flower graphs

2007
*
International Mathematical Forum
*

We define a class

doi:10.12988/imf.2007.07221
fatcat:jabdw4nibnbh3pjx6xvtje3hyq
*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*. ...##
###
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]

2006
*
Lecture Notes in Computer Science
*

We show that the

doi:10.1007/11917496_18
fatcat:wx5yb77xd5g2dgvd3lufnhkdqy
*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. ...
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