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A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics and Symmetry/Asymmetry".

Deadline for manuscript submissions: 31 March 2023 | Viewed by 3629

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Special Issue Editors

Dr. Andrea Semaničová-Feňovčíková

E-Mail Website
Guest Editor

Department of Applied Mathematics and Informatics, Faculty of Mechanical Engineering, Technical University in Košice, Košice, Slovakia
Interests: graph labelings; metric dimension of graphs
Special Issues, Collections and Topics in MDPI journals

Prof. Dr. Martin Bača

E-Mail Website
Guest Editor

Special Issue Information

Dear Colleagues,

Some of the central topics in graph theory are that of graph labeling and graph coloring. The graph labeling/coloring problem involves assigning labels/colors to certain set of graph elements subject to certain restrictions and constraints. Both graph labelings and graph colorings can be used to solve a wide variety of problems in real world, as well as theoretical challenges.

The distance between two vertices, i.e., the length of a shortest path between these vertices, is the basis of the definition of many graph parameters including metric dimension. The metric dimension and its variants have appeared in various applications of graph theory.

Please note that all submitted papers must be within the general scope of the Symmetry journal.

Dr. Andrea Semaničová-Feňovčíková
Prof. Dr. Martin Bača
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2000 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

graceful labelings and their variations
magic-type labelings
antimagic-type labelings
irregular-type labelings
sum labelings and their variations
prime and vertex prime labelings
binary labelings
average labelings
labelings and their induced colorings
applications of graph labelings
vertex colorings
edge colorings
face and map coloring
list coloring
path coloring
total coloring
applications of graph colorings
metric dimension
strong metric dimension
local metric dimension
adjacency dimension
k-metric dimension
partition dimension and its variants
fractional metric dimension and its variants

Published Papers (6 papers)

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Research

Open AccessArticle

On Rotationally Symmetrical Planar Networks and Their Local Fractional Metric Dimension

Shahbaz Ali

Rashad Ismail

Francis Joseph H. Campena

Hanen Karamti

and

Muhammad Usman Ghani

Symmetry 2023, 15(2), 530; https://doi.org/10.3390/sym15020530 - 16 Feb 2023

Cited by 2 | Viewed by 459

Abstract

The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a map

ϑ : V (G) \to [0, 1]

such that

\sum_{u \in R {v, w}} ϑ (u) \geq 1,

for every pair of adjacent distinct vertices

v, w \in V (G)

. The local fractional metric dimension of the graph G is defined as

{ldim}_{f} (G)

min {\sum_{v \in V (G)} ϑ (v)

, where

ϑ

is a local resolving function of

G}

. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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Open AccessArticle

The Ascending Ramsey Index of a Graph

Gary Chartrand

and

Ping Zhang

Symmetry 2023, 15(2), 523; https://doi.org/10.3390/sym15020523 - 15 Feb 2023

Viewed by 431

Abstract

Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence

G_{1}

G_{2}

, …,

G_{k}

of pairwise edge-disjoint [...] Read more.

Let G be a graph with a given red-blue coloring c of the edges of G. An ascending Ramsey sequence in G with respect to c is a sequence

G_{1}

G_{2}

, …,

G_{k}

of pairwise edge-disjoint subgraphs of G such that each subgraph

G_{i}

(

1 \leq i \leq k

) is monochromatic and

G_{i}

is isomorphic to a proper subgraph of

G_{i + 1}

(

1 \leq i \leq k - 1

). The ascending Ramsey index

A R_{c} (G)

of G with respect to c is the maximum length of an ascending Ramsey sequence in G with respect to c. The ascending Ramsey index

A R (G)

of G is the minimum value of

A R_{c} (G)

among all red-blue colorings c of G. It is shown that there is a connection between this concept and set partitions. The ascending Ramsey index is investigated for some classes of highly symmetric graphs such as complete graphs, matchings, stars, graphs consisting of a matching and a star, and certain double stars. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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Open AccessArticle

On the P₃ Coloring of Graphs

Hong Yang

Muhammad Naeem

and

Shahid Qaisar

Symmetry 2023, 15(2), 521; https://doi.org/10.3390/sym15020521 - 15 Feb 2023

Viewed by 317

Abstract

The vertex coloring of graphs is a well-known coloring of graphs. In this coloring, all of the vertices are assigned colors in such a way that no two adjacent vertices have the same color. We can call this type of coloring

P_{2}

[...] Read more.

P_{2}

coloring, where

P_{2}

is a path graph. However, there are situations in which this type of coloring cannot give us the solution to the problem at hand. To answer such questions, in this article, we introduce a novel graph coloring called

P_{3}

coloring. A graph is called

P_{3}

-colorable if we can assign colors to the vertices of the graph such that the vertices of every

P_{3}

path are distinct. The minimum number of colors required for a graph to have

P_{3}

coloring is called the

P_{3}

chromatic number. The aim of this article is, in general, to prove some basic results concerning this coloring, and, in particular, to compute the

P_{3}

chromatic number for different symmetric families of graphs. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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Open AccessArticle

Modular Version of Edge Irregularity Strength for Fan and Wheel Graphs

Debi Oktia Haryeni

Zata Yumni Awanis

Martin Bača

and

Andrea Semaničová-Feňovčíková

Symmetry 2022, 14(12), 2671; https://doi.org/10.3390/sym14122671 - 16 Dec 2022

Cited by 1 | Viewed by 472

Abstract

A k-labeling from the vertex set of a simple graph

G = (V, E)

to a set of integers

{1, 2, \dots, k}

is defined to be a modular edge irregular if, for every [...] Read more.

A k-labeling from the vertex set of a simple graph

G = (V, E)

to a set of integers

{1, 2, \dots, k}

is defined to be a modular edge irregular if, for every couple of distinct edges, their modular edge weights are distinct. The modular edge weight is the remainder of the division of the sum of end vertex labels by modulo

| E (G) |

. The modular edge irregularity strength of a graph is known as the maximal vertex label k, minimized over all modular edge irregular k-labelings of the graph. In this paper we describe labeling schemes with symmetrical distribution of even and odd edge weights and investigate the existence of (modular) edge irregular labelings of joins of paths and cycles with isolated vertices. We estimate the bounds of the (modular) edge irregularity strength for the join graphs

P_{n} + \bar{K_{m}}

and

C_{n} + \bar{K_{m}}

and determine the corresponding exact value of the (modular) edge irregularity strength for some fan graphs and wheel graphs in order to prove the sharpness of the presented bounds. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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Open AccessArticle

Generalized Arithmetic Staircase Graphs and Their Total Edge Irregularity Strengths

and

Symmetry 2022, 14(9), 1853; https://doi.org/10.3390/sym14091853 - 06 Sep 2022

Viewed by 554

Abstract

Let

Γ = (V_{Γ}, E_{Γ})

be a simple undirected graph with finite vertex set

V_{Γ}

and edge set

E_{Γ}

. A total

n

-labeling [...] Read more.

Let

Γ = (V_{Γ}, E_{Γ})

be a simple undirected graph with finite vertex set

V_{Γ}

and edge set

E_{Γ}

. A total

n

-labeling

α : V_{Γ} \cup E_{Γ} \to {1, 2, \dots, n}

is called a total edge irregular labeling on

Γ

if for any two different edges

x y

and

x^{'} y^{'}

E_{Γ}

the numbers

α (x) + α (x y) + α (y)

and

α (x^{'}) + α (x^{'} y^{'}) + α (y^{'})

are distinct. The smallest positive integer n such that

Γ

can be labeled by a total edge irregular labeling is called the total edge irregularity strength of the graph

Γ

. In this paper, we provide the total edge irregularity strength of some asymmetric graphs and some symmetric graphs, namely generalized arithmetic staircase graphs and generalized double-staircase graphs, as the generalized forms of some existing staircase graphs. Moreover, we give the construction of the corresponding total edge irregular labelings. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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Open AccessArticle

Distance Antimagic Product Graphs

Rinovia Simanjuntak

and

Aholiab Tritama

Symmetry 2022, 14(7), 1411; https://doi.org/10.3390/sym14071411 - 09 Jul 2022

Viewed by 597

Abstract

A distance antimagic graph is a graph G admitting a bijection

f : V (G) \to {1, 2, \dots, | V (G) |}

such that for two distinct vertices x and y, [...] Read more.

A distance antimagic graph is a graph G admitting a bijection

f : V (G) \to {1, 2, \dots, | V (G) |}

such that for two distinct vertices x and y,

ω (x) \neq ω (y)

, where

ω (x) = \sum_{y \in N (x)} f (y)

, for

N (x)

the open neighborhood of x. It was conjectured that a graph G is distance antimagic if and only if G contains no two vertices with the same open neighborhood. In this paper, we study several distance antimagic product graphs. The products under consideration are the three fundamental graph products (Cartesian, strong, direct), the lexicographic product, and the corona product. We investigate the consequence of the non-commutative (or sometimes called non-symmetric) property of the last two products to the antimagicness of the product graphs. Full article

(This article belongs to the Special Issue Labelings, Colorings and Distances in Graphs)

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