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Open AccessArticle

The Difference of Zagreb Indices of Halin Graphs

Lina Zheng

Yiqiao Wang

and

Weifan Wang

Axioms 2023, 12(5), 450; https://doi.org/10.3390/axioms12050450 - 02 May 2023

Viewed by 545

Abstract

The difference of Zagreb indices of a graph G is defined as [...] Read more.

The difference of Zagreb indices of a graph G is defined as

Δ M (G) = \sum_{u \in V (G)} {(d (u))}^{2} - \sum_{u v \in E (G)} d (u) d (v)

, where

d (x)

denotes the degree of a vertex x in G. A Halin graph G is a graph that results from a plane tree T without vertices of degree two and with at least one vertex of degree at least three such that all leaves are joined through a cycle C in the embedded order. In this paper, we establish both lower and upper bounds on the difference of Zagreb indices for general Halin graphs and some special Halin graphs with fewer inner vertices. Furthermore, extremal graphs attaining related bounds are found. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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Open AccessArticle

Graphs with Strong Proper Connection Numbers and Large Cliques

Yingbin Ma

Xiaoxue Zhang

and

Yanfeng Xue

Axioms 2023, 12(4), 353; https://doi.org/10.3390/axioms12040353 - 03 Apr 2023

Viewed by 650

Abstract

In this paper, we mainly investigate graphs with a small (strong) proper connection number and a large clique number. First, we discuss the (strong) proper connection number of a graph G of order n and

ω (G) = n - i

[...] Read more.

ω (G) = n - i

for

1 ⩽ i ⩽ 3

. Next, we investigate the rainbow connection number of a graph G of order n,

d i a m (G) \geq 3

and

ω (G) = n - i

for

2 ⩽ i ⩽ 3

. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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Open AccessArticle

On Several Parameters of Super Line Graph

L_{2} (G)

Jiawei Meng

Baoyindureng Wu

and

Hongliang Ma

Axioms 2023, 12(3), 276; https://doi.org/10.3390/axioms12030276 - 06 Mar 2023

Viewed by 847

Abstract

The super line graph of index r, denoted by

L_{r} (G)

, is defined for any graph G with at least r edges. Its vertices are the sets of r edges of G, and two such sets are [...] Read more.

The super line graph of index r, denoted by

L_{r} (G)

, is defined for any graph G with at least r edges. Its vertices are the sets of r edges of G, and two such sets are adjacent if an edge of one is adjacent to an edge of the other. In this paper, we give an explicit characterization for all graphs G with

L_{2} (G)

being a complete graph. We present lower bounds for the clique number and chromatic number of

L_{2} (G)

for several classes of graphs. In addition, bounds for the domination number of

L_{2} (G)

are established in terms of the domination number of the line graph

L (G)

of a graph. A number of related problems on

L_{2} (G)

are proposed for a further study. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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Open AccessArticle

Extremal Graphs for Sombor Index with Given Parameters

Wanping Zhang

Jixiang Meng

and

Na Wang

Axioms 2023, 12(2), 203; https://doi.org/10.3390/axioms12020203 - 15 Feb 2023

Viewed by 812

Abstract

In this paper, we present the upper and lower bounds on Sombor index

S O (G)

among all connected graphs (respectively, connected bipartite graphs). We give some sharp lower and upper bounds on

S O (G)

among connected graphs [...] Read more.

In this paper, we present the upper and lower bounds on Sombor index

S O (G)

among all connected graphs (respectively, connected bipartite graphs). We give some sharp lower and upper bounds on

S O (G)

among connected graphs in terms of some parameters, including chromatic, girth and matching number. Meanwhile, we characterize the extremal graphs attaining those bounds. In addition, we give upper bounds on

S O (G)

among connected bipartite graphs with given matching number and/or connectivity and determine the corresponding extremal connected bipartite graphs. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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Open AccessArticle

The Decomposability of Toroidal Graphs without Adjacent Triangles or Short Cycles

Huajing Lu

and

Fengwei Li

Axioms 2023, 12(2), 173; https://doi.org/10.3390/axioms12020173 - 08 Feb 2023

Cited by 1 | Viewed by 736

Abstract

A graph G has a

(d, h)

-decomposition if there is a pair

(D, F)

such that F is a subgraph of G and D is an acyclic orientation of

G - E (F)

, [...] Read more.

A graph G has a

(d, h)

-decomposition if there is a pair

(D, F)

such that F is a subgraph of G and D is an acyclic orientation of

G - E (F)

, where the maximum degree of F is no more than h and the maximum out-degree of D is no more than d. This paper proves that toroidal graphs having no adjacent triangles are

(3, 1)

-decomposable, and for

{i, j} \subseteq {3, 4, 6}

, toroidal graphs without i- and j-cycles are

(2, 1)

-decomposable. As consequences of these results, toroidal graphs without adjacent triangles are 1-defective DP-4-colorable, and toroidal graphs without i- and j-cycles are 1-defective DP-3-colorable for

{i, j} \subseteq {3, 4, 6}

. Full article

(This article belongs to the Special Issue Graph Theory and Discrete Applied Mathematics)

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