Extended Chromatic Polynomials

1972 ◽  
Vol 24 (3) ◽  
pp. 492-501 ◽  
Author(s):  
Andrew Sobczyk ◽  
James O. Gettys
Keyword(s):  
Positive Integer ◽  
Finite Graph ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Vertex Set

Let G be a finite graph with non-empty vertex set (G) and edge set (G) (see [2]). Let λ be a positive integer. Tutte [5] defines a λ-colouring of G as a mapping of (G) into the set Iλ = {1, 2, 3, …, λ} with the property that two ends of any edge are mapped onto distinct integers. The elements of Iλ are commonly called “colours.” If P(G, λ) represents the number of λ-colourings of G, it is well known that P(G, λ) can be expressed as a polynomial in λ. For this reason P(G, λ) is usually referred to as the chromatic polynomial of G.The chromatic polynomial P(G, λ) was first suggested as an approach to the four-colour conjecture. To quote Tutte [5]: ”… many people are specially interested in the value λ = 4.

scholarly journals Regions Without Complex Zeros for Chromatic Polynomials on Graphs with Bounded Degree

2008 ◽  
Vol 17 (2) ◽  
pp. 225-238 ◽  
Author(s):  
ROBERTO FERNÁNDEZ ◽  
ALDO PROCACCI
Keyword(s):  
Finite Graph ◽  
Chromatic Polynomial ◽  
Maximal Degree ◽  
Bounded Degree ◽  
Chromatic Polynomials ◽  
Image Position ◽  
Complex Zeros

We prove that the chromatic polynomial$P_\mathbb{G}(q)$of a finite graph$\mathbb{G}$of maximal degree Δ is free of zeros for |q| ≥C*(Δ) withThis improves results by Sokal and Borgs. Furthermore, we present a strengthening of this condition for graphs with no triangle-free vertices.

scholarly journals Fuzzy Chromatic Polynomial of Fuzzy Graphs with Crisp and Fuzzy Vertices Using α-Cuts

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Mamo Abebe Ashebo ◽  
V. N. Srinivasa Rao Repalle
Keyword(s):  
Optimization Problems ◽  
Real Life ◽  
Chromatic Polynomial ◽  
Fuzzy Graph ◽  
Traffic Light ◽  
Chromatic Polynomials ◽  
Combinatorial Optimization Problems ◽  
Fuzzy Graphs ◽  
Vertex Set ◽  
Fuzzy Vertex

Coloring of fuzzy graphs has many real life applications in combinatorial optimization problems like traffic light system, exam scheduling, register allocation, etc. In this paper, the concept of fuzzy chromatic polynomial of fuzzy graph is introduced and defined based on α-cuts of fuzzy graph. Two different types of fuzziness to fuzzy graph are considered in the paper. The first type was fuzzy graph with crisp vertex set and fuzzy edge set and the second type was fuzzy graph with fuzzy vertex set and fuzzy edge set. Depending on this, the fuzzy chromatic polynomials for some fuzzy graphs are discussed. Some interesting remarks on fuzzy chromatic polynomial of fuzzy graphs have been derived. Further, some results related to the concept are proved. Lastly, fuzzy chromatic polynomials for complete fuzzy graphs and fuzzy cycles are studied and some results are obtained.

scholarly journals Chromatic Bounds on Orbital Chromatic Roots

10.37236/4412 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Dae Hyun Kim ◽  
Alexander H. Mun ◽  
Mohamed Omar
Keyword(s):  
Positive Integer ◽  
Real Root ◽  
Chromatic Polynomial ◽  
Outerplanar Graphs ◽  
Chromatic Polynomials ◽  
The Real ◽  
Real Roots ◽  
Chromatic Roots

Given a group $G$ of automorphisms of a graph $\Gamma$, the orbital chromatic polynomial $OP_{\Gamma,G}(x)$ is the polynomial whose value at a positive integer $k$ is the number of orbits of $G$ on proper $k$-colorings of $\Gamma.$ Cameron and Kayibi introduced this polynomial as a means of understanding roots of chromatic polynomials. In this light, they posed a problem asking whether the real roots of the orbital chromatic polynomial of any graph are bounded above by the largest real root of its chromatic polynomial. We resolve this problem in a resounding negative by not only constructing a counterexample, but by providing a process for generating families of counterexamples. We additionally begin the program of finding classes of graphs whose orbital chromatic polynomials have real roots bounded above by the largest real root of their chromatic polynomials; in particular establishing this for many outerplanar graphs.

scholarly journals Zero-Free Intervals of Chromatic Polynomials of Mixed Hypergraphs

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 193
Author(s):  
Ruixue Zhang ◽  
Fengming Dong ◽  
Meiqiao Zhang
Keyword(s):  
Positive Integer ◽  
Chromatic Polynomial ◽  
Chromatic Polynomials ◽  
Finite Set ◽  
Graph Function ◽  
Mixed Hypergraph

A mixed hypergraph H is a triple (X,C,D), where X is a finite set and each of C and D is a family of subsets of X. For any positive integer λ, a proper λ-coloring of H is an assignment of λ colors to vertices in H such that each member in C contains at least two vertices assigned the same color and each member in D contains at least two vertices assigned different colors. The chromatic polynomial of H is the graph-function counting the number of distinct proper λ-colorings of H whenever λ is a positive integer. In this article, we show that chromatic polynomials of mixed hypergraphs under certain conditions are zero-free in the intervals (−∞,0) and (0,1), which extends known results on zero-free intervals of chromatic polynomials of graphs and hypergraphs.

scholarly journals Bipartite graphs with close domination and k-domination numbers

2020 ◽  
Vol 18 (1) ◽  
pp. 873-885
Author(s):  
Gülnaz Boruzanlı Ekinci ◽  
Csilla Bujtás
Keyword(s):  
Positive Integer ◽  
Dominating Set ◽  
Domination Number ◽  
Bipartite Graphs ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Necessary And Sufficient ◽  
The Given

Abstract Let k be a positive integer and let G be a graph with vertex set V(G) . A subset D\subseteq V(G) is a k -dominating set if every vertex outside D is adjacent to at least k vertices in D . The k -domination number {\gamma }_{k}(G) is the minimum cardinality of a k -dominating set in G . For any graph G , we know that {\gamma }_{k}(G)\ge \gamma (G)+k-2 where \text{Δ}(G)\ge k\ge 2 and this bound is sharp for every k\ge 2 . In this paper, we characterize bipartite graphs satisfying the equality for k\ge 3 and present a necessary and sufficient condition for a bipartite graph to satisfy the equality hereditarily when k=3 . We also prove that the problem of deciding whether a graph satisfies the given equality is NP-hard in general.

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

Random flights on regular graphs

1984 ◽  
Vol 16 (03) ◽  
pp. 618-637 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Automorphism Group ◽  
Transition Probabilities ◽  
Finite Graph ◽  
Initial Position ◽  
Regular Graphs ◽  
Random Flights ◽  
The Past ◽  
Initial Location ◽  
Vertex Set

Let K be a finite graph with vertex set V = {x 0, x 1, …, xσ –1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v 0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn –1 = xi }, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

scholarly journals On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
I. H. Agustin ◽  
M. I. Utoyo ◽  
Dafik ◽  
M. Venkatachalam ◽  
Surahmat
Keyword(s):  
Natural Number ◽  
Finite Graph ◽  
Star Graph ◽  
Pendant Vertex ◽  
Vertex Set

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

THE ITERATION DIGRAPHS OF GROUP RINGS OVER FINITE FIELDS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350162 ◽  
Author(s):  
YANGJIANG WEI ◽  
GAOHUA TANG ◽  
JIZHU NAN
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Directed Edge ◽  
Cycle Structure ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cn over a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCn and k such that G(𝔽prCn, k) is semiregular.

scholarly journals The radius of unit graphs of rings

2021 ◽  
Vol 6 (10) ◽  
pp. 11508-11515
Author(s):  
Zhiqun Li ◽  
◽  
Huadong Su
Keyword(s):  
Positive Integer ◽  
Simple Graph ◽  
Unit Graph ◽  
Vertex Set ◽  
Ring Extensions ◽  
Nonzero Identity

<abstract><p>Let $ R $ be a ring with nonzero identity. The unit graph of $ R $ is a simple graph whose vertex set is $ R $ itself and two distinct vertices are adjacent if and only if their sum is a unit of $ R $. In this paper, we study the radius of unit graphs of rings. We prove that there exists a ring $ R $ such that the radius of unit graph can be any given positive integer. We also prove that the radius of unit graphs of self-injective rings are 1, 2, 3, $ \infty $. We classify all self-injective rings via the radius of its unit graph. The radius of unit graphs of some ring extensions are also considered.</p></abstract>

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