Symmetry

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13 pages, 267 KiB

Open AccessArticle

Plane Partitions and Divisors

by Cristina Ballantine and Mircea Merca

Symmetry 2024, 16(1), 5; https://0-doi-org.brum.beds.ac.uk/10.3390/sym16010005 - 19 Dec 2023

Viewed by 833

Abstract

In this paper, we consider the sum of divisors d of n such that

n / d

is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum [...] Read more.

In this paper, we consider the sum of divisors d of n such that

n / d

is a power of 2 and derive a new decomposition for the number of plane partitions of n in terms of binomial coefficients as a sum over partitions of n. In this context, we introduce a new combinatorial interpretation of the number of plane partitions of n. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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14 pages, 340 KiB

Open AccessArticle

Labeling Circulant Graphs: Distance Two Condition

by Laila Loudiki, Ez-Zobair Bidine and Mustapha Kchikech

Symmetry 2023, 15(12), 2160; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15122160 - 05 Dec 2023

Viewed by 713

Abstract

Given

n \geq 6

D = {1, 2, \dots, ⌊ \frac{n}{2} ⌋}

, and a generating set

S \subseteq D

, the circulant graph

C_{n} (S)

has

Z_{n}

as a vertex set [...] Read more.

Given

n \geq 6

D = {1, 2, \dots, ⌊ \frac{n}{2} ⌋}

, and a generating set

S \subseteq D

, the circulant graph

C_{n} (S)

has

Z_{n}

as a vertex set in which two distinct vertices i and j are adjacent if and only if

{| i - j |}_{n} \in S

, where

{| x |}_{n} = min (| x |, n - | x |)

is the circular distance modulo n. In this paper, we determine the

L (2, 1)

-labeling number of

C_{n} (D ∖ X)

, referred to as

λ (C_{n} (D ∖ X))

, for

X = {⌊ \frac{n}{2} ⌋}

X = {a}

X = {a, b}

, and in the general case when

| X | < ⌊ \frac{n}{2} ⌋ - ⌈ \frac{n}{4} ⌉

, where

a, b \in D

. Furthermore, we demonstrate that for all

n \geq 6

and any given set S,

λ (C_{n} (S)) = n + g c d (n, \bar{S}) - 2

if and only if

g c d (n, \bar{S}) \geq 2

, and

λ (C_{n} (S)) \leq n - 1

if and only if

g c d (n, \bar{S}) = 1

. Additionally, we establish that when the diameter of

C_{n} (S)

equals 2,

λ (C_{n} (S)) = n - 1

. This observation motivated us to investigate the properties of S that lead to a diameter of

C_{n} (S)

equal to 2. Then, we introduce a highly distinctive family, denoted as

A_{n}

, that generates a large number of generating sets. For each value of n, we acquire a circulant graph

C_{n} (A_{n})

with a diameter of 2,

λ (C_{n} (A_{n})) = n - 1

, and various additional interesting properties. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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20 pages, 315 KiB

Open AccessArticle

n-Color Partitions into Distinct Parts as Sums over Partitions

by Mircea Merca and Emil Simion

Symmetry 2023, 15(11), 2067; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15112067 - 15 Nov 2023

Viewed by 983

Abstract

The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For

r \in {0, 1}

, let

Q L_{r} (n)

be the number of n-color [...] Read more.

The partitions in which the parts of size n can come in n different colors are known as n-color partitions. For

r \in {0, 1}

, let

Q L_{r} (n)

be the number of n-color partitions of n into distinct parts which have a number of parts congruent to r modulo 2. In this paper, we consider specializations of complete and elementary symmetric functions in order to establish two kinds of formulas for

Q L_{0} (n) \pm Q L_{1} (n)

as sums over partitions of n in terms of binomial coefficients. The first kind of formulas only involve partitions in which the parts of size n appear at most n times, while the second kind of formulas involve unrestricted partitions. Similar results are obtained for the first differences of

Q L_{0} (n) \pm Q L_{1} (n)

and the partial sums of

Q L_{0} (n) \pm Q L_{1} (n)

. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

16 pages, 315 KiB

Open AccessArticle

Statistical Analyses of a Class of Random Cyclooctatetraene Chain Networks with Respect to Several Topological Properties

by Chen Tao, Shengjun Tang and Xianya Geng

Symmetry 2023, 15(11), 1971; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15111971 - 24 Oct 2023

Viewed by 634

Abstract

In recent years, the research on complex networks has created a boom. The objective of the present paper is to study a random cyclooctatetraene chain whose graph-theoretic mathematical properties arose scientists’ interests. By applying the concept of symmetry and probability theory, we obtain the explicit analytical expressions for the variances of Schultz index, multiplicative degree-Kirchhoff index Gutman index, and additive degree-Kirchhoff index of a random cyclooctatetraene chain with n octagons, which plays a crucial role in the research and application of topological indices. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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14 pages, 306 KiB

Open AccessArticle

The λ-Fold Spectrum Problem for the Orientations of the Eight-Cycle

by Şafak Durukan-Odabaşı and Uğur Odabaşı

Symmetry 2023, 15(10), 1930; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15101930 - 18 Oct 2023

Viewed by 1104

Abstract

A D-decomposition of a graph (or digraph) G is a partition of the edge set (or arc set) of G into subsets, where each subset induces a copy of the fixed graph D. Graph decomposition finds motivation in numerous practical applications, particularly in the realm of symmetric graphs, where these decompositions illuminate intricate symmetrical patterns within the graph, aiding in various fields such as network design, and combinatorial mathematics, among various others. Of particular interest is the case where G is

^{λ} K_{v}^{*}

, the

λ

-fold complete symmetric digraph on v vertices, that is, the digraph with

λ

directed edges in each direction between each pair of vertices. For a given digraph D, the set of all values v for which

^{λ} K_{v}^{*}

has a D-decomposition is called the

λ

-fold spectrum of D. An eight-cycle has 22 non-isomorphic orientations. The

λ

-fold spectrum problem has been solved for one of these oriented cycles. In this paper, we provide a complete solution to the

λ

-fold spectrum problem for each of the remaining 21 orientations. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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8 pages, 231 KiB

Open AccessArticle

Inexact Iterates of Nonexpansive Mappings with Summable Errors in Metric Spaces with Graphs

by Simeon Reich and Alexander J. Zaslavski

Symmetry 2023, 15(10), 1927; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15101927 - 17 Oct 2023

Viewed by 790

Abstract

In our joint work with Dan Butnariu (2006) we established the stability of the convergence of iterates of a nonexpansive mapping on a complete metric space in the presence of summable computational errors. In a recent paper of ours, we extended this result to inexact iterates of nonexpansive mappings on complete metric spaces with graphs under a certain assumption on the iterates. In the present paper we obtain an analogous result by removing that assumption on the iterates and replacing it with an additional assumption on the graph. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

17 pages, 743 KiB

Open AccessArticle

On Laplacian Eigenvalues of Wheel Graphs

by Manal Alotaibi, Ahmad Alghamdi and Hanan Alolaiyan

Symmetry 2023, 15(9), 1737; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091737 - 11 Sep 2023

Viewed by 910

Abstract

Consider G to be a simple graph with n vertices and m edges, and

L (G)

to be a Laplacian matrix with Laplacian eigenvalues of

μ_{1}, μ_{2}, \dots, μ_{n} = z e r o

. [...] Read more.

Consider G to be a simple graph with n vertices and m edges, and

L (G)

to be a Laplacian matrix with Laplacian eigenvalues of

μ_{1}, μ_{2}, \dots, μ_{n} = z e r o

. Write

S_{k} (G) = \sum_{i = 1}^{k} μ_{i}

as the sum of the k-largest Laplacian eigenvalues of G, where

k \in {1, 2, \dots, n}

. The motivation of this study is to solve a conjecture in algebraic graph theory for a special type of graph called a wheel graph. Brouwer’s conjecture states that

S_{k} (G) \leq m + (\binom{k + 1}{2})

, where

k = 1, 2, \dots, n

. This paper proves Brouwer’s conjecture for wheel graphs. It also provides an upper bound for the sum of the largest Laplacian eigenvalues for the wheel graph

W_{n + 1},

which provides a better approximation for this upper bound using Brouwer’s conjecture and the Grone–Merris–Bai inequality. We study the symmetry of wheel graphs and recall an example of the symmetry group of

W_{n + 1}

n \geq 3

. We obtain our results using majorization methods and illustrate our findings in tables, diagrams, and curves. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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15 pages, 780 KiB

Open AccessArticle

Total Perfect Roman Domination

by Ahlam Almulhim

Symmetry 2023, 15(9), 1676; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15091676 - 31 Aug 2023

Cited by 1 | Viewed by 692

Abstract

A total perfect Roman dominating function (TPRDF) on a graph

G = (V, E)

is a function f from V to

{0, 1, 2}

satisfying (i) every vertex v with

f (v) = 0

[...] Read more.

A total perfect Roman dominating function (TPRDF) on a graph

G = (V, E)

is a function f from V to

{0, 1, 2}

satisfying (i) every vertex v with

f (v) = 0

is a neighbor of exactly one vertex u with

f (u) = 2

; in addition, (ii) the subgraph of G that is induced by the vertices with nonzero weight has no isolated vertex. The weight of a TPRDF f is

\sum_{v \in V} f (v)

. The total perfect Roman domination number of G, denoted by

γ_{t R}^{p} (G)

, is the minimum weight of a TPRDF on G. In this paper, we initiated the study of total perfect Roman domination. We characterized graphs with the largest-possible

γ_{t R}^{p} (G)

. We proved that total perfect Roman domination is NP-complete for chordal graphs, bipartite graphs, and for planar bipartite graphs. Finally, we related

γ_{t R}^{p} (G)

to perfect domination

γ^{p} (G)

by proving

γ_{t R}^{p} (G) \leq 3 γ^{p} (G)

for every graph G, and we characterized trees T of order

n \geq 3

for which

γ_{t R}^{p} (T) = 3 γ^{p} (T)

. This notion can be utilized to develop a defensive strategy with some properties. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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12 pages, 444 KiB

Open AccessArticle

Convex Polygon Triangulation Based on Symmetry

by Predrag V. Krtolica, Predrag S. Stanimirović, Sead H. Mašović, Islam A. Elshaarawy and Alena Stupina

Symmetry 2023, 15(8), 1526; https://0-doi-org.brum.beds.ac.uk/10.3390/sym15081526 - 02 Aug 2023

Viewed by 884

Abstract

A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal

δ_{i, n}

and diagonal

δ_{n - i, n}

, [...] Read more.

A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal

δ_{i, n}

and diagonal

δ_{n - i, n}

i \in {2, \dots, ⌊ n / 2 ⌋}

results in mirror images. Obviously, there are

⌊ n / 2 ⌋ - 1

pairs of these reflectively symmetrical images. The influence of the observed symmetry on polygon triangulation is studied. The central result of this research is the construction of an efficient algorithm used for generating convex polygon triangulations in minimal time and without generating repeat triangulations. The proposed algorithm uses the diagonal values of the Catalan triangle to avoid duplicate triangulations with negligible computational costs and provides significant speedups compared to known methods. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory)

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