A note on global dominator coloring of graphs

Rangarajan, R.; Kalarkop, David A.

doi:10.1142/s1793830921501585

Cited by 2 publications

(9 citation statements)

References 5 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The Apostol–Euler polynomials of the first kind

{scriptE}_{n} false(y, λ false)

are defined by means of the following generating function:

f_{e} false(w, y; λ false) = \frac{e^{w y}}{λ e^{w} + 1} = true \sum_{n = 0}^{\infty} \frac{{scriptE}_{n} false(y; λ false)}{2 n!} w^{n},

(cf. previous studies 4–38 ).…”

Section: Introductionmentioning

confidence: 56%

“…The Stirling numbers of the second kind are also given by the following formula:

y^{v} = true \sum_{d = 0}^{v} S_{2} false(v, d false) y_{false(d false)},

(cf. previous studies 1–39 ; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 76%

“…Substituting

y = 0

into (), we have the Apostol–Euler numbers:

E_{n} (λ) = E_{n} (0, λ) .

Putting

y = 0

and

λ = 1

in (), we have the Euler numbers:

E_{n} = E_{n}^{(1)} (1) = E_{n} (0) .

(cf. previous studies 4–38 ; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 76%

“…Recently, Rubinstein 30 gave an extension of the Bernoulli and Eulerian polynomials of higher order with the aid of vector partition function. On the subject of the generating functions for special polynomials and numbers of higher order with their many different extensions, a remarkably large number of investigations have found in the literature (see, e.g., previous studies 1–39 ).…”

Section: Special Numbers and Polynomials Arising From The Generating ...mentioning

confidence: 99%

“…For details about the w coloring of a graph G, the chromatic functions, and the chromatic polynomials, see previous studies. 5,16,17 By using the above observations and comments, many properties of the w coloring of a graph G (colored with no more than w colors), the chromatic functions, the chromatic polynomials, the famous problems in mathematics related to the four-color theorem, other generating functions, and combinatoric problems can be studied and investigated.…”

Section: Minimal Polynomial and Polynomials Of The Chordal Graphmentioning

confidence: 99%

See 4 more Smart Citations

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

Şimşek

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

The aim of this article is to construct some new families of generating‐type functions interpolating a certain class of higher order Bernoulli‐type, Euler‐type, Apostol‐type numbers, and polynomials. Applying the umbral calculus convention method and the shift operator to these functions, these generating functions are investigated in many different aspects such as applications related to the finite calculus, combinatorial analysis, the chordal graph, number theory, and complex analysis especially partial fraction decomposition of rational functions associated with Laurent expansion. By using the falling factorial function and the Stirling numbers of the first kind, we also construct new families of generating functions for certain classes of higher order Apostol‐type numbers and polynomials, the Bernoulli numbers and polynomials, the Fubini numbers, and others. Many different relations among these generating functions, difference equation including the Eulerian numbers, the shift operator, minimal polynomial, polynomial of the chordal graph, and other applications are given. Moreover, further remarks and comments on the results of this paper are presented.

show abstract

“…The Apostol–Euler polynomials of the first kind

{scriptE}_{n} false(y, λ false)

are defined by means of the following generating function:

f_{e} false(w, y; λ false) = \frac{e^{w y}}{λ e^{w} + 1} = true \sum_{n = 0}^{\infty} \frac{{scriptE}_{n} false(y; λ false)}{2 n!} w^{n},

(cf. previous studies 4–38 ).…”

Section: Introductionmentioning

confidence: 56%

“…The Stirling numbers of the second kind are also given by the following formula:

y^{v} = true \sum_{d = 0}^{v} S_{2} false(v, d false) y_{false(d false)},

(cf. previous studies 1–39 ; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 76%

“…Substituting

y = 0

into (), we have the Apostol–Euler numbers:

E_{n} (λ) = E_{n} (0, λ) .

Putting

y = 0

and

λ = 1

in (), we have the Euler numbers:

E_{n} = E_{n}^{(1)} (1) = E_{n} (0) .

(cf. previous studies 4–38 ; see also the references cited in each of these earlier works).…”

Section: Introductionmentioning

confidence: 76%

Section: Special Numbers and Polynomials Arising From The Generating ...mentioning

confidence: 99%

Section: Minimal Polynomial and Polynomials Of The Chordal Graphmentioning

confidence: 99%

See 3 more Smart Citations

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

Şimşek

2021

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

An Extensive Study on Graph Colourings and Dominator Chromatic Number of Sugeno-Type Fuzzy Graphs

Devi

2022

Mathematical Problems in Engineering

View full text Add to dashboard Cite

The concept of graph colouring has become a very active field of research that enhances many practical applications and theoretical challenges. An accurate (vertex) colouring through the condition that each vertex is whichever unaccompanied in its colour class or adjoining to all vertices of at least one colour class is known as dominator colouring. A graph is dominator colouring is a suitable colouring in which at least one vertex dominates each colour class. Many difficult and global issues are solved using fuzzy graph colouring approaches. In this paper, we introduce a new Sugeno-Type Fuzzy graph of groups, which is found using several fuzzy graph operations such as dominant path-colouring number, cycle, union, join, and products. A number that is representative based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths is the minimum number of shared edges based on those vertices in the formations of all paths with those vertices as their starts and ends to compare with other paths. The sugeno dominant path-colouring number is the minimum Sugeno number of shared edges among the Sugeno cardinality of all sets of shared edges, which allows for various techniques. These results are used in studying various and recently introduced chromatic numbers of graphs.

show abstract

A note on global dominator coloring of graphs

Cited by 2 publications

References 5 publications

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

Applications of constructed new families of generating‐type functions interpolating new and known classes of polynomials and numbers

An Extensive Study on Graph Colourings and Dominator Chromatic Number of Sugeno-Type Fuzzy Graphs

Contact Info

Product

Resources

About