Research

10 pages, 265 KiB

Open AccessArticle

The Hamilton–Waterloo Problem with C₁₆-Factors and C_m-Factors for Odd m

by Li Wang

Symmetry 2024, 16(3), 371; https://doi.org/10.3390/sym16030371 - 19 Mar 2024

Viewed by 519

The Hamilton–Waterloo problem is a problem of graph factorization. The Hamilton–Waterloo problem HWP

(H; m, n; α, β)

asks for a two-factorization of a graph H containing

α

C_{m}

-factors and

β

C_{n}

[...] Read more.

The Hamilton–Waterloo problem is a problem of graph factorization. The Hamilton–Waterloo problem HWP

(H; m, n; α, β)

asks for a two-factorization of a graph H containing

α

C_{m}

-factors and

β

C_{n}

-factors. Let

K_{v}^{*}

denote the complete graph

K_{v}

if v is odd and

K_{v}

minus a one-factor if v is even. In this paper, we completely solve the Hamilton–Waterloo problem HWP

(K_{v}^{*}; m, 16; α, β)

for odd

m \geq 9

and

α \geq 15

. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

16 pages, 484 KiB

Open AccessArticle

The Maximum Clique Problem and Integer Programming Models, Their Modifications, Complexity and Implementation

by Milos Seda

Symmetry 2023, 15(11), 1979; https://doi.org/10.3390/sym15111979 - 26 Oct 2023

Cited by 1 | Viewed by 1348

Abstract

The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional [...] Read more.

The maximum clique problem is a problem that takes many forms in optimization and related graph theory problems, and also has many applications. Because of its NP-completeness (nondeterministic polynomial time), the question arises of its solvability for larger instances. Instead of the traditional approaches based on the use of approximate or stochastic heuristic methods, we focus here on the use of integer programming models in the GAMS (General Algebraic Modelling System) environment, which is based on exact methods and sophisticated deterministic heuristics incorporated in it. We propose modifications of integer models, derive their time complexities and show their direct use in GAMS. GAMS makes it possible to find optimal solutions to the maximum clique problem for instances with hundreds of vertices and thousands of edges within minutes at most. For extremely large instances, good approximations of the optimum are given in a reasonable amount of time. A great advantage of this approach over all the mentioned algorithms is that even if GAMS does not find the best known solution within the chosen time limit, it displays its value at the end of the calculation as a reachable bound. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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10 pages, 302 KiB

Open AccessArticle

The Cross-Intersecting Family of Certain Permutation Groups

by Hua Han

Symmetry 2023, 15(9), 1708; https://doi.org/10.3390/sym15091708 - 06 Sep 2023

Viewed by 558

Abstract

Two subsets X and Y of a permutation group G acting on

Ω

are cross-intersecting if for every

x \in X

and every

y \in Y

there exists some point

α \in Ω

such that

α^{x} = α^{y}

. Based on [...] Read more.

Two subsets X and Y of a permutation group G acting on

Ω

are cross-intersecting if for every

x \in X

and every

y \in Y

there exists some point

α \in Ω

such that

α^{x} = α^{y}

. Based on several observations made on the cross-independent version of Hoffman’s theorem, we characterize in this paper the cross-intersecting families of certain permutation groups. Our proof uses a Cayley graph on a permutation subgroup with respect to the derangement. By carefully analyzing the cross-independent version of Hoffman’s theorem, we obtain a useful theorem to consider cross-intersecting subsets of certain kinds of permutation subgroups, such as

PGL (2, q)

,

PSL (2, q)

and

S_{n}

. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

17 pages, 496 KiB

Open AccessArticle

On the Cube Polynomials of Padovan and Lucas–Padovan Cubes

by Gwangyeon Lee and Jinsoo Kim

Symmetry 2023, 15(7), 1389; https://doi.org/10.3390/sym15071389 - 10 Jul 2023

Viewed by 686

Abstract

The hypercube is one of the best models for the network topology of a distributed system. Recently, Padovan cubes and Lucas–Padovan cubes have been introduced as new interconnection topologies. Despite their asymmetric and relatively sparse interconnections, the Padovan and Lucas–Padovan cubes are shown [...] Read more.

The hypercube is one of the best models for the network topology of a distributed system. Recently, Padovan cubes and Lucas–Padovan cubes have been introduced as new interconnection topologies. Despite their asymmetric and relatively sparse interconnections, the Padovan and Lucas–Padovan cubes are shown to possess attractive recurrent structures. In this paper, we determine the cube polynomial of Padovan cubes and Lucas–Padovan cubes, as well as the generating functions for the sequences of these cubes. Several explicit formulas for the coefficients of these polynomials are obtained, in particular, they can be expressed with convolved Padovan numbers and Lucas–Padovan numbers. In particular, the coefficients of the cube polynomials represent the number of hypercubes, a symmetry inherent in Padovan and Lucas–Padovan cubes. Therefore, cube polynomials are very important for characterizing these cubes. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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14 pages, 324 KiB

Open AccessArticle

An Extension of Sylvester’s Theorem on Arithmetic Progressions

by Augustine O. Munagi and Francisco Javier de Vega

Symmetry 2023, 15(6), 1276; https://doi.org/10.3390/sym15061276 - 18 Jun 2023

Cited by 1 | Viewed by 1180

Abstract

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of [...] Read more.

Sylvester’s theorem states that every number can be decomposed into a sum of consecutive positive integers except powers of 2. In a way, this theorem characterizes the partitions of a number as a sum of consecutive integers. The first generalization we propose of the theorem characterizes the partitions of a number as a sum of arithmetic progressions with positive terms. In addition to synthesizing and rediscovering known results, the method we propose allows us to state a second generalization and characterize the partitions of a number into parts whose differences between consecutive parts form an arithmetic progression. To achieve this, we will analyze the set of divisors in arithmetics that modify the usual definition of the multiplication operation between two integers. As we will see, symmetries arise in the set of divisors based on two parameters:

t_{1}

, being even or odd, and

t_{2}

, congruent to 0, 1, or 2 (mod 3). This approach also leads to a unique representation result of the same nature as Sylvester’s theorem, i.e., a power of 3 cannot be represented as a sum of three or more terms of a positive integer sequence such that the differences between consecutive terms are consecutive integers. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

14 pages, 363 KiB

Open AccessArticle

[k]-Roman Domination in Digraphs

by Xinhong Zhang, Xin Song and Ruijuan Li

Symmetry 2023, 15(3), 743; https://doi.org/10.3390/sym15030743 - 17 Mar 2023

Viewed by 853

Abstract

Let

D = (V (D), A (D))

be a finite, simple digraph and k a positive integer. A function [...] Read more.

Let

D = (V (D), A (D))

be a finite, simple digraph and k a positive integer. A function

f : V (D) \to {0, 1, 2, \dots, k + 1}

is called a

[k]

-Roman dominating function (for short,

[k]

-RDF) if

f (A N^{-} [v]) \geq | A N^{-} (v) | + k

for any vertex

v \in V (D)

, where

A N^{-} (v) = {u \in N^{-} (v) : f (u) \geq 1}

and

A N^{-} [v] = A N^{-} (v) \cup {v}

. The weight of a

[k]

-RDF f is

ω (f) = \sum_{v \in V (D)} f (v)

. The minimum weight of any

[k]

-RDF on

D

is the

[k]

-Roman domination number, denoted by

γ_{[k R]} (D)

. For

k = 2

and

k = 3

, we call them the double Roman domination number and the triple Roman domination number, respectively. In this paper, we presented some general bounds and the Nordhaus–Gaddum bound on the

[k]

-Roman domination number and we also determined the bounds on the

[k]

-Roman domination number related to other domination parameters, such as domination number and signed domination number. Additionally, we give the exact values of

γ_{[k R]} (P_{n})

and

γ_{[k R]} (C_{n})

for the directed path

P_{n}

and directed cycle

C_{n}

. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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10 pages, 1582 KiB

Open AccessArticle

On Non-Zero Vertex Signed Domination

by Baogen Xu, Mengmeng Zheng and Ting Lan

Symmetry 2023, 15(3), 741; https://doi.org/10.3390/sym15030741 - 17 Mar 2023

Viewed by 882

Abstract

For a graph

G = (V, E)

and a function

f : V \to {- 1, + 1}

, if

S \subseteq V

then we write [...] Read more.

For a graph

G = (V, E)

and a function

f : V \to {- 1, + 1}

, if

S \subseteq V

then we write

f (S) = \sum_{v \in S} f (v)

. A function

f

is said to be a non-zero vertex signed dominating function (for short, NVSDF) of

G

if

f (N [v]) = 0

holds for every vertex

v

in

G

, and the non-zero vertex signed domination number of

G

is defined as

γ_{s b} (G) = \max {f (V) | f is an NVSDF of G} .

In this paper, the novel concept of the non-zero vertex signed domination for graphs is introduced. There is also a special symmetry concept in graphs. Some upper bounds of the non-zero vertex signed domination number of a graph are given. The exact value of

γ_{s b} (G)

for several special classes of graphs is determined. Finally, we pose some open problems. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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14 pages, 688 KiB

Open AccessArticle

Applications on Topological Indices of Zero-Divisor Graph Associated with Commutative Rings

by Clement Johnson Rayer and Ravi Sankar Jeyaraj

Symmetry 2023, 15(2), 335; https://doi.org/10.3390/sym15020335 - 25 Jan 2023

Cited by 4 | Viewed by 1633

Abstract

A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. Let

R

be a commutative ring with identity, and

Z^{*} (R)

is [...] Read more.

A topological index is a numeric quantity associated with a chemical structure that attempts to link the chemical structure to various physicochemical properties, chemical reactivity, or biological activity. Let

R

be a commutative ring with identity, and

Z^{*} (R)

is the set of all non-zero zero divisors of

R

. Then,

Γ (R)

is said to be a zero-divisor graph if and only if

a \cdot b = 0

, where

a, b \in V (Γ (R)) = Z^{*} (R)

and

(a, b) \in E (Γ (R))

. We define

a \sim b

if

a \cdot b = 0

or

a = b

. Then, ∼ is always reflexive and symmetric, but ∼ is usually not transitive. Then,

Γ (R)

is a symmetric structure measured by the ∼ in commutative rings. Here, we will draw the zero-divisor graph from commutative rings and discuss topological indices for a zero-divisor graph by vertex eccentricity. In this paper, we will compute the total eccentricity index, eccentric connectivity index, connective eccentric index, eccentricity based on the first and second Zagreb indices, Ediz eccentric connectivity index, and augmented eccentric connectivity index for the zero-divisor graph associated with commutative rings. These will help us understand the characteristics of various symmetric physical structures of finite commutative rings. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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13 pages, 324 KiB

Open AccessArticle

The Singularity of Four Kinds of Tricyclic Graphs

by Haicheng Ma, Shang Gao and Bin Zhang

Symmetry 2022, 14(12), 2507; https://doi.org/10.3390/sym14122507 - 28 Nov 2022

Viewed by 1271

Abstract

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of [...] Read more.

A singular graph G, defined when its adjacency matrix is singular, has important applications in mathematics, natural sciences and engineering. The chemical importance of singular graphs lies in the fact that if the molecular graph is singular, the nullity (the number of the zero eigenvalue) is greater than 0, then the corresponding chemical compound is highly reactive or unstable. By this reasoning, chemists have a great interest in this problem. Thus, the problem of characterization singular graphs was proposed and raised extensive studies on this challenging problem thereafter. The graph obtained by conglutinating the starting vertices of three paths

P_{s_{1}}

,

P_{s_{2}}

,

P_{s_{3}}

into a vertex, and three end vertices into a vertex on the cycle

C_{a_{1}}

,

C_{a_{2}}

,

C_{a_{3}}

, respectively, is denoted as

γ (a_{1}, a_{2}, a_{3}, s_{1}, s_{2}, s_{3})

. Note that

δ (a_{1}, a_{2}, a_{3}, s_{1}, s_{2}) = γ (a_{1}, a_{2}, a_{3}, s_{1}, 1, s_{2})

,

ζ (a_{1}, a_{2}, a_{3}, s) = γ (a_{1}, a_{2}, a_{3}, 1, 1, s)

,

φ (a_{1}, a_{2}, a_{3}) = γ (a_{1}, a_{2}, a_{3}, 1, 1, 1)

. In this paper, we give the necessity and sufficiency that the

γ -

graph,

δ -

graph,

ζ -

graph and

φ -

graph are singular and prove that the probability that a randomly given

γ -

graph,

δ -

graph,

ζ -

graph or

φ -

graph being singular is equal to

\frac{325}{512}, \frac{165}{256}, \frac{43}{64}

,

\frac{21}{32}

, respectively. From our main results, we can conclude that such a

γ -

graph(

δ -

graph,

ζ -

graph,

φ -

graph) is singular if at least one cycle is a multiple of 4 in length, and surprisingly, the theoretical probability of these graphs being singular is more than half. This result promotes the understanding of a singular graph and may be promising to propel the solutions to relevant application problems. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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25 pages, 1777 KiB

Open AccessArticle

On Neighborhood Inverse Sum Indeg Energy of Molecular Graphs

by Sourav Mondal, Biswajit Some, Anita Pal and Kinkar Chandra Das

Symmetry 2022, 14(10), 2147; https://doi.org/10.3390/sym14102147 - 14 Oct 2022

Cited by 7 | Viewed by 1161

Abstract

The spectral graph theory explores connections between combinatorial features of graphs and algebraic properties of associated matrices. The neighborhood inverse sum indeg (

N I

) index was recently proposed and explored to be a significant molecular descriptor. Our aim is to investigate [...] Read more.

The spectral graph theory explores connections between combinatorial features of graphs and algebraic properties of associated matrices. The neighborhood inverse sum indeg (

N I

) index was recently proposed and explored to be a significant molecular descriptor. Our aim is to investigate the

N I

index from a spectral standpoint, for which a suitable matrix is proposed. The matrix is symmetric since it is generated from the edge connection information of undirected graphs. A novel graph energy is introduced based on the eigenvalues of that matrix. The usefulness of the energy as a molecular structural descriptor is analyzed by investigating predictive potential and isomer discrimination ability. Fundamental mathematical properties of the present spectrum and energy are investigated. The spectrum of the bipartite class of graphs is identified to be symmetric about the origin of the real line. Bounds of the spectral radius and the energy are explained by identifying the respective extremal graphs. Full article

(This article belongs to the Special Issue Advances in Combinatorics and Graph Theory)

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