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44 pages, 1327 KB

Open AccessFeature PaperArticle

by Igal Sason, Noam Krupnik, Suleiman Hamud and Abraham Berman

Mathematics 2025, 13(4), 549; https://doi.org/10.3390/math13040549 - 7 Feb 2025

Cited by 1 | Viewed by 1830

The study of spectral graph determination is a fascinating area of research in spectral graph theory and algebraic combinatorics. This field focuses on examining the spectral characterization of various classes of graphs, developing methods to construct or distinguish cospectral nonisomorphic graphs, and analyzing the conditions under which a graph’s spectrum uniquely determines its structure. This paper presents an overview of both classical and recent advancements in these topics, along with newly obtained proofs of some existing results, which offer additional insights. Full article

(This article belongs to the Special Issue Graph Theory and Applications, 2nd Edition)

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9 pages, 233 KB

Open AccessArticle

One Turán Type Problem on Uniform Hypergraphs

by Linlin Wang and Sujuan Liu

Axioms 2024, 13(8), 544; https://doi.org/10.3390/axioms13080544 - 11 Aug 2024

Viewed by 847

Abstract

Let

n, m, p, r \in N

with

p \geq n \geq r

. For a hypergraph, if each edge has r vertices, then the hypergraph is called an r-graph. Define [...] Read more.

Let

n, m, p, r \in N

with

p \geq n \geq r

. For a hypergraph, if each edge has r vertices, then the hypergraph is called an r-graph. Define

e_{r} (n, m; p)

to be the maximum number of edges of an r-graph with p vertices in which every subgraph of n vertices has at most m edges. Researching this function constitutes a Turán type problem. In this paper, on the one hand, for fixed p, we present some results about the exact values of

e_{r} (n, m; p)

for small m compared to n; on the other hand, for sufficient large p, we use the combinatorial technique of double counting to give an upper bound of

e (n, m; p)

and obtain a lower bound of

e_{r} (n, m; p)

by applying the lower bound of the independent set of a hypergraph. Full article

17 pages, 322 KB

Open AccessArticle

BDAC: Boundary-Driven Approximations of K-Cliques

by Büşra Çalmaz and Belgin Ergenç Bostanoğlu

Symmetry 2024, 16(8), 983; https://doi.org/10.3390/sym16080983 - 2 Aug 2024

Cited by 1 | Viewed by 1468

Abstract

Clique counts are crucial in applications like detecting communities in social networks and recurring patterns in bioinformatics. Counting k-cliques—a fully connected subgraph with k nodes, where each node has a direct, mutual, and symmetric relationship with every other node—becomes computationally challenging for larger k due to combinatorial explosion, especially in large, dense graphs. Existing exact methods have difficulties beyond k = 10, especially on large datasets, while sampling-based approaches often involve trade-offs in terms of accuracy, resource utilization, and efficiency. This difficulty becomes more pronounced in dense graphs as the number of potential k-cliques grows exponentially. We present Boundary-driven approximations of k-cliques (BDAC), a novel algorithm that approximates k-clique counts without using recursive procedures or sampling methods. BDAC offers both lower and upper bounds for k-cliques at local (per-vertex) and global levels, making it ideal for large, dense graphs. Unlike other approaches, BDAC’s complexity remains unaffected by the value of k. We demonstrate its effectiveness by comparing it with leading algorithms across various datasets, focusing on k values ranging from 8 to 50. Full article

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

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9 pages, 254 KB

Open AccessArticle

A Class of Fibonacci Matrices, Graphs, and Games

by Valentin E. Brimkov and Reneta P. Barneva

Mathematics 2022, 10(21), 4038; https://doi.org/10.3390/math10214038 - 31 Oct 2022

Cited by 1 | Viewed by 1797

Abstract

In this paper, we define a class of Fibonacci graphs as graphs whose adjacency matrices are obtained by alternating binary Fibonacci words. We show that Fibonacci graphs are close in size to Turán graphs and that their size-stability tradeoff defined as the product of their size and stability number is very close to the maximum possible over all bipartite graphs. We also consider a combinatorial game based on sequential vertex deletions and show that the Fibonacci graphs are extremal regarding the number of rounds in which the game can terminate. Full article

(This article belongs to the Section E: Applied Mathematics)

18 pages, 343 KB

Open AccessArticle

On General Reduced Second Zagreb Index of Graphs

by Lkhagva Buyantogtokh, Batmend Horoldagva and Kinkar Chandra Das

Mathematics 2022, 10(19), 3553; https://doi.org/10.3390/math10193553 - 29 Sep 2022

Cited by 4 | Viewed by 2258

Abstract

Graph-based molecular structure descriptors (often called “topological indices”) are useful for modeling the physical and chemical properties of molecules, designing pharmacologically active compounds, detecting environmentally hazardous substances, etc. The graph invariant

G R M_{α}

, known under the name general reduced second [...] Read more.

G R M_{α}

, known under the name general reduced second Zagreb index, is defined as

G R M_{α} (Γ) = \sum_{u v \in E (Γ)} (d_{Γ} (u) + α) (d_{Γ} (v) + α)

, where

d_{Γ} (v)

is the degree of the vertex v of the graph

Γ

and

α

is any real number. In this paper, among all trees of order n, and all unicyclic graphs of order n with girth g, we characterize the extremal graphs with respect to

G R M_{α}

(α \geq - \frac{1}{2})

. Using the extremal unicyclic graphs, we obtain a lower bound on

G R M_{α} (Γ)

of graphs in terms of order n with k cut edges, and completely determine the corresponding extremal graphs. Moreover, we obtain several upper bounds on

G R M_{α}

of different classes of graphs in terms of order n, size m, independence number

γ

, chromatic number k, etc. In particular, we present an upper bound on

G R M_{α}

of connected triangle-free graph of order

n > 2

m > 0

edges with

α > - 1.5

, and characterize the extremal graphs. Finally, we prove that the Turán graph

T_{n} (k)

gives the maximum

G R M_{α} (α \geq - 1)

among all graphs of order n with chromatic number k. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

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9 pages, 321 KB

Open AccessArticle

A Note on the Lagrangian of Linear 3-Uniform Hypergraphs

by Sinan Hu and Biao Wu

Symmetry 2022, 14(7), 1402; https://doi.org/10.3390/sym14071402 - 7 Jul 2022

Cited by 2 | Viewed by 2098

Abstract

Lots of symmetric properties are well-explored and analyzed in extremal graph theory, such as the well-known symmetrization operation in the Turán problem and the high symmetric in the extremal graphs. This paper is devoted to studying the Lagrangian of hypergraphs, which connects to [...] Read more.

π_{λ} (F)

is the limit supremum of

r! λ (G)

over all F-free G, where

λ (G)

is the Lagrangian of G. An r-uniform hypergraph F is called

λ

-perfect if

π_{λ} (F)

equals

r! λ (K_{v (F) - 1}^{r})

. Yan and Peng conjectured that: for integer

r \geq 3

, there exists

n_{0} (r)

such that if G and H are two

λ

-perfect r-graphs with

| V (G) |

and

| V (H) |

no less than

n_{0} (r)

, then the disjoint union of G and H is

λ

-perfect. Let

S_{t}

denote a 3-uniform hypergraph with t edges

{e_{1}, \dots, e_{t}}

satisfying that

e_{i} \cap e_{j} = {v}

for all

1 \leq i < j \leq t

. In this paper, we show that the conjecture holds for

G = S_{2}

and

H = S_{t}

for all

t \geq 62

. Moreover, our result yields a class of Turán densities of 3-uniform hypergraphs. In our proof, we use some new techniques to study Lagrangian density problems; using a result of Sidorenko to find subgraphs, and a result of Talbot to upper bound the Lagrangian of a hypergraph. Full article

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

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