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Open AccessArticle

The Generalized Characteristic Polynomial of the K_m,n-Complement of a Bipartite Graph

by Weiliang Zhao and Helin Gong

Symmetry 2025, 17(3), 328; https://doi.org/10.3390/sym17030328 - 21 Feb 2025

Viewed by 807

The generalized matrix of a graph G is defined as

M (G) = A (G) - t D (G)

(

t \in R

, and

A (G)

and

D (G)

, respectively, denote [...] Read more.

The generalized matrix of a graph G is defined as

M (G) = A (G) - t D (G)

(

t \in R

, and

A (G)

and

D (G)

, respectively, denote the adjacency matrix and the degree matrix of G), and the generalized characteristic polynomial of G is merely the characteristic polynomial of

M (G)

. Let

K_{m, n}

be the complete bipartite graph. Then, the

K_{m, n}

-complement of a subgraph G in

K_{m, n}

is defined as the graph obtained by removing all edges of an isomorphic copy of G from

K_{m, n}

. In this paper, by using a determinant expansion on the sum of two matrices (one of which is a diagonal matrix), a general method for computing the generalized characteristic polynomial of the

K_{m, n}

-complement of a bipartite subgraph G is provided. Furthermore, when G is a graph with rank no more than 4, the explicit formula for the generalized characteristic polynomial of the

K_{m, n}

-complements of G is given. Full article

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

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9 pages, 255 KB

Open AccessArticle

The Strong Equitable Vertex 1-Arboricity of Complete Bipartite Graphs and Balanced Complete k-Partite Graphs

by Janejira Laomala, Keaitsuda Maneeruk Nakprasit, Kittikorn Nakprasit and Watcharintorn Ruksasakchai

Symmetry 2022, 14(2), 287; https://doi.org/10.3390/sym14020287 - 31 Jan 2022

Viewed by 2266

Abstract

An equitable k-coloring of a graph G is a proper k-coloring of G such that the sizes of any two color classes differ by at most one. An equitable $(q, r)$ -tree-coloring of a graph G is an equitable q-coloring of G such that the subgraph induced by each color class is a forest of maximum degree at most r. Let the strong equitable vertex r-arboricity of a graph

G,

denoted by

v a_{r}^{\equiv} (G)

, be the minimum p such that G has an equitable

(q, r)

-tree-coloring for every

q \geq p .

The values of

v a_{1}^{\equiv} (K_{n, n})

were investigated by Tao and Lin and Wu, Zhang, and Li where exact values of

v a_{1}^{\equiv} (K_{n, n})

were found in some special cases. In this paper, we extend their results by giving the exact values of

v a_{1}^{\equiv} (K_{n, n})

for all cases. In the process, we introduce a new function related to an equitable coloring and obtain a more general result by determining the exact value of each

v a_{1}^{\equiv} (K_{m, n})

and

v a_{1}^{\equiv} (G)

where G is a balanced complete k-partite graph

K_{n, \dots, n}

. Both complete bipartite graphs

K_{m, n}

and balanced complete k-partite graphs

K_{n, \dots, n}

are symmetry in several aspects and also studied broadly. For the other aspect of symmetry, by the definition of equitable k-coloring of graphs, in a specific case that the number of colors divides the number of vertices of graph, we can say that the graph is a balanced k-partite graph. Full article

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

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