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Advances in Combinatorics, Discrete Mathematics and Graph Theory

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Dr. Jonnathan Rodriguez

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Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Av Angamos 601, Antofagasta, Chile
Interests: discrete mathematics; graph theory; matrix theory; combinatorics; group theory; graphs; matrix; algebra

Dr. Luis Medina

E-Mail Website
Guest Editor

Departamento de Matemáticas, Facultad de Ciencias Básicas, Universidad de Antofagasta, Av Angamos 601, Antofagasta, Chile
Interests: discrete mathematics; graph theory; matrix theory; linear algebra
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Special Issue Information

Dear Colleagues,

Combinatorics and graph theory are areas of mathematics that deal with the study of discrete objects and their combinatorial structures. In recent decades, there have been significant advances in these areas, driven by a combination of new theoretical approaches, advanced computational tools, and applications in various fields. Regarding graph theory, significant progress has been made in areas such as random graph theory, complex network theory, graph coloring theory, graph flow theory, and structural graph theory. For example, in random graph theory, profound results have been obtained on the asymptotic properties of random graphs, leading to a better understanding of random phenomena in complex real-world networks, such as social networks, communication networks, and biological networks. In addition to these theoretical advances, combinatorics and graph theory have also experienced a surge in their application in various interdisciplinary areas, such as computer science, computational biology, information theory, game theory, and data science.

Dr. Jonnathan Rodriguez
Dr. Luis Medina
Guest Editors

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Published Papers (3 papers)

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Research

9 pages, 337 KiB

Open AccessArticle

Graphs with a Fixed Maximum Degree and Order Attaining the Upper Bound on Minimum Status

by Wei-Han Tsai, Jen-Ling Shang and Chiang Lin

Mathematics 2024, 12(22), 3600; https://doi.org/10.3390/math12223600 - 17 Nov 2024

Viewed by 562

Abstract

The status (or transmission) of a vertex in a connected graph is the sum of distances between the vertex and all other vertices. The minimum status (or minimum transmission) of a connected graph is the minimum of the statuses of all vertices in the graph. Previously, sharp lower and upper bounds have been obtained on the minimum status of connected graphs with a fixed maximum degree k and order n. Moreover, for

2 \leq k \leq \frac{n}{2}

, the following theorem about graphs attaining the maximum on the minimum status has also been proposed without proof. The theorem is as follows: Let G be a connected graph of order n with

▵ (G) = k

, where

2 \leq k \leq \frac{n}{2}

. Then, the minimum status of G attains the maximum if and only if one of the following holds. (1) G is a path or a cycle, where

k = 2

; (2)

G_{k, n}

is a spanning subgraph of G and G is a spanning subgraph of

H_{k, n}

, where

3 \leq k < \frac{n}{2}

; and (3) either

G_{\frac{n}{2}, n}

is a spanning subgraph of G and G is a spanning subgraph of

H_{\frac{n}{2}, n}

G_{\frac{n}{2}, n}

is a spanning subgraph of G and G is a spanning subgraph of

H_{n}

, where

k = \frac{n}{2}

for even

n \geq 6

. For the integers

n, k

with

2 \leq k \leq n - 1

, the graph

G_{k, n}

has the vertex set

V (G_{k, n}) = {x_{1}, x_{2}, \dots, x_{n}}

and the edge set

E (G_{k, n}) = {x_{i} x_{i + 1} : i = 1, 2, \dots, n - k} \cup {x_{n - k + 1} x_{j} : j = n - k + 2, n - k + 3, \dots, n}

; the graph

H_{k, n}

is obtained from

G_{k, n}

by adding all the edges

x_{i} x_{j}

, where

n - k + 2 \leq i < j \leq n

; and for even

n \geq 6

the graph

H_{n}

is obtained from

G_{\frac{n}{2}, n}

by adding the edge

x_{\frac{n}{2} - 1} x_{\frac{n}{2} + 2}

and all the edges

x_{i} x_{j}

, where

\frac{n}{2} + 3 \leq i < j \leq n

. This study provides the proof to complete the above theorem. Full article

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

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12 pages, 485 KiB

Open AccessArticle

Conjectures About Wheels Without One Edge with Paths and Cycles

by Michal Staš and Mária Timková

Mathematics 2024, 12(22), 3484; https://doi.org/10.3390/math12223484 - 7 Nov 2024

Viewed by 458

Abstract

The crossing number

cr (G)

of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of this paper is to give the crossing numbers of the join products [...] Read more.

The crossing number

cr (G)

of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main aim of this paper is to give the crossing numbers of the join products

G^{*} + P_{n}

and

G^{*} + C_{n}

for the connected graph

G^{*}

obtained by removing one edge (incident with the dominating vertex) from the wheel

W_{5}

on six vertices, and where

P_{n}

and

C_{n}

are paths and cycles on n vertices, respectively. Finally, we also introduce four new conjectures concerning crossing numbers of the join products of

P_{n}

and

C_{n}

with

W_{m} ∖ e

obtained by removing one edge (of both possible types) from the wheel

W_{m}

m + 1

vertices. Full article

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

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Figure 1

12 pages, 283 KiB

Open AccessArticle

Algebraic Connectivity of Power Graphs of Finite Cyclic Groups

by Bilal Ahmad Rather

Mathematics 2024, 12(14), 2175; https://doi.org/10.3390/math12142175 - 11 Jul 2024

Cited by 1 | Viewed by 660

Abstract

The power graph

P (Z_{n})

Z_{n}

for a finite cyclic group

Z_{n}

is a simple undirected connected graph such that two distinct nodes

x

and

y

Z_{n}

are adjacent in

P (Z_{n})

[...] Read more.

The power graph

P (Z_{n})

Z_{n}

for a finite cyclic group

Z_{n}

is a simple undirected connected graph such that two distinct nodes

x

and

y

Z_{n}

are adjacent in

P (Z_{n})

if and only if

x \neq y

and

x^{i} = y

y^{i} = x

for some non-negative integer

i

. In this article, we find the Laplacian eigenvalues of

P (Z_{n})

and show that

P (Z_{n})

is Laplacian integral (integer algebraic connectivity) if and only if

n

is either the product of two distinct primes or a prime power. That answers a conjecture by Panda, Graphs and Combinatorics, (2019). Full article

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

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