Complete Bipartite Graph Research Articles

Integer k-matching preclusion of graphs

As a generalization of matching preclusion number of a graph, we provide the (strong) integer $k$-matching preclusion number, abbreviated as $MP^{k}$ number ($SMP^{k}$ number), which is the minimum number of edges (vertices and edges) whose deletion results in a graph that has neither perfect integer $k$-matching nor almost perfect integer $k$-matching. In this paper, we obtain a necessary condition of graphs with an almost-perfect integer $k$-matching and a relational expression between the matching number and the integer $k$-matching number of bipartite graphs. And then the $MP^{k}$ number and the $SMP^{k}$ number of complete graphs, bipartite graphs and arrangement graphs are obtained, respectively.

A Spectral Extremal Problem on Non-Bipartite Triangle-Free Graphs

A theorem of Nosal and Nikiforov states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda(G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. A well-known spectral conjecture of Bollobás and Nikiforov [J. Combin. Theory Ser. B 97 (2007)] asserts that if $G$ is a $K_{r+1}$-free graph with $m$ edges, then $\lambda_1^2(G) + \lambda_2^2(G) \le (1-\frac{1}{r})2m$. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] confirmed the conjecture in the case $r=2$. Using this base case, they proved further that $\lambda (G)\le \sqrt{m-1}$ for every non-bipartite triangle-free graph $G$, with equality if and only if $m=5$ and $G=C_5$. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented an improvement by showing $\lambda (G) \le \beta (m)$, where $\beta(m)$ is the largest root of $Z(x):=x^3-x^2-(m-2)x+m-3$. The equality in Zhai--Shu's result holds only if $m$ is odd and $G$ is obtained from the complete bipartite graph $K_{2,\frac{m-1}{2}}$ by subdividing exactly one edge. Motivated by this observation, Zhai and Shu proposed a question to find a sharp bound when $m$ is even. We shall solve this question by using a different method and characterize three kinds of spectral extremal graphs over all triangle-free non-bipartite graphs with even size. Our proof technique is mainly based on applying Cauchy's interlacing theorem of eigenvalues of a graph, and with the aid of a triangle counting lemma in terms of both eigenvalues and the size of a graph.

Spectral Radius of Graphs with Given Size and Odd Girth

Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m,k)>\sqrt{m-k+3}$ such that every non bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m,k)$ must contain an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of the complete bipartite graph $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of Zhai and Shu [Discrete Math. 345 (2022)] and settles a conjecture of Li and Peng [The Electronic J. Combin. 29 (4) (2022)] as well.

Quasistationary Distribution for The Invasion Model on a Complete Bipartite Graph

The Invasion Model on the complete bipartite graph was introduced and studied by physicists as a rudimentary model for opinion dynamics on complex networks. Our main result provides the limit of the Quasistationary Distribution (QSD) for the model as one partition size tends to infinity. This limit is a highly dispersed measure. The results are in stark contrast to the Voter Model whose QSD is essentially a perturbation to absorbing states. A distinct feature not present in the Voter Model and demanding a new approach developed here is the appearance of two time scales with a non-trivial interaction. Our analysis is based on decoupling of the scales and using a Stein-type operator to identify a limit distribution. It also involves an application of a known duality with coalescing Markov chains through which asymptotic survival rates associated with the QSD are obtained.

Product of Semi – Lattices of Certain Graphs

Introduction: In this article, author tries to construct a relation between graphs of product of meet-semilattices L = L_1 X L_2, where L_1 and L_2 are two semilattices and obtain some properties of such graphs. Author investigated that for meet-semilattices L_1 and L_2 has a cycle of length n-1 and n.
 Objectives: author reveals that if L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2, then it is a star graph. In this paper, we have covered some definitions, examples and theorems on zero devisor graph edge of a 4 - cycles or a 5 - cycles. Γ(L) is a star graph.
 Methods: Theorem 1.1 [9] The zero-divisor graph of a finite meet-semilattice with only one atom is the empty graph. The zero-divisor graph of the meet semilattice is the empty graph. However, this does not hold for infinite meet-semilattices with one atom. For consider, the infinite meet-semilattice, where the descending dots represent infinite descending chain. It has only one atom c but its graph Γ(L) is an infinite star graph.
 Theorem 1.2 [12] Every disconnected graph cannot be a graph of any meet- semilattice L with 0.
 Remark 1.2 A graphs of product of meet-semilattices and obtain some properties of such graphs. In this section, we consider two integral meet-semilattices L_1 and L_2 with L ∼=L_1 X L_2 and show that if ∣L_1∣ = m+1, ∣L_2∣ = n+1, then Γ(L) is the complete bipartite graph K (m, n).
 Results: If L does not contain any atom, then any edge in Γ(L) is contained in a cycle of length ≤ 6, and therefore Γ(L) is a union of 4 - cycles and 5 - cycles. Let L be a meet-semilattice with 0. If Γ(L) contains a cycle, then the core K of Γ(L) is a union of 4- cycles and 5 – cycles and any vertex in Γ(L) is either a vertex of the core K of Γ(L) or is a pendant of Γ(L). Let L_1 and L_2 be two meet-semilattices with 0 and L = L_1 X L_2. Then exactly one of the following holds:
 
 Γ(L) has a cycle of length n-1 or n (that is gr Γ(L)≤ n).
 Γ(L) is a star graph.
 
 Conclusions: In this article we have studied the concept of the zero-divisor graph derived from meet-semilattice L with 0 on the lines of Anderson and Livingston [6]. Also, we generalized certain results from Demeyer, Mckenzie and Schneider [18] to meet-semilattice L with 0.

A Note to Non-adaptive Broadcasting

Broadcasting is a fundamental problem in the information dissemination area. In classical broadcasting, a message must be sent from one network member to all other members as rapidly as feasible. Although it has been demonstrated that this problem is NP-Hard for arbitrary graphs, it has several applications in various fields. As a result, the universal lists model, replicating real-world restrictions like the memory limits of nodes in large networks, is introduced as a branch of this problem in the literature. In the universal lists model, each node is equipped with a fixed list and has to follow the list regardless of the originator. In this study, we focus on the non-adaptive branch of universal lists broadcasting. In this regard, we establish the optimal broadcast time of [Formula: see text]-ary trees and binomial trees under the non-adaptive model and provide an upper bound for complete bipartite graphs. We also improved a general upper bound for trees under the same model and showed that our upper bound cannot be improved in general.

Some Results on Odd-Even Congruence Labeling of Graphs

Introduction: Labeling of graphs has been introduced in 1966. Assignment of natural numbers to vertices and/or edges is referred as graph labeling. Inspired by the ample application of graph labeling technique in real life problems, multifarious labeling strategy was adopted and investigated by many researchers.
 Objectives:Graph labeling plays a vital role in various fields and can be implemented in multitudinous discipline including coding theory, X-ray, Psychology, crystallography, circuit design, communication networks, astronomy, radar, data security, secret sharing, data base management and so on. Apart from these labeling techniques serve as a model to understand discrete mathematical domains.
 Methodology:In this paper, an attempt has been made to introduce new labeling such as odd-even congruence labeling. Congruence Graph Labeling is an allocation of natural numbers as labels for the edges and vertices of a graph based on modular arithmetic property. Odd-even congruence labeling is an allocation of odd integers to vertices and even integers to edges in addition to congruence graph labeling.
 Result: The suggested labeling has been identified on complete bipartite graph, comb graph and spliting graph of a star graph. Further, it is proved that graph acquired by connecting two copies of even cycle Cr by a path Pt, K1,t⊗P2 and D2 (Pt) are odd-even congurence graph.

Degree based energy and spectral radius of a graph with self-loops

Let GX be a graph obtained from a simple graph G by attaching a self-loop at each vertex of X ⊆ V ( G ) . The general extended adjacency matrix for the graph GX is defined and the bounds for the degree based energy of the graph GX are obtained. The study extends the notion of degree based energy of simple graphs to graphs with self-loops. For the graph GX of order n and size m with σ self-loops, the adjacency energy, E ( G X ) ≥ 4 mn + 2 σ ( n − σ ) n . The spectral radius ρ ( G X ) of its adjacency matrix is always less than or equal to 1 2 [ − 1 + 1 + 4 ( 2 m + σ + Δ − 1 ) ] , where Δ is the maximum degree in the graph GX and the equality conditions are given for 0 < σ < n . Few more bounds for ρ ( G X ) are also obtained. The study shows that, the spectral radius f max ( G X ) of its extended adjacency matrix satisfies: min i ∼ j F ij ( 2 m + σ ) n ≤ f max ( G X ) ≤ max i ∼ j F ij 2 ( 2 m + σ ) . We conclude the article by computing the extended adjacency spectrum of complete graph and complete bipartite graphs with self-loops.

Minimum degree stability of C2k+1 ${C}_{2k+1}$‐free graphs

AbstractWe consider the minimum degree stability of graphs forbidding odd cycles: What is the tight bound on the minimum degree to guarantee that the structure of a ‐free graph inherits from the extremal graph (a balanced complete bipartite graph)? Andrásfai, Erdős, and Sós showed that if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist showed that for , if a ‐free graph on vertices has minimum degree greater than , then it is bipartite. Häggkvist also pointed out that this result cannot be extended to . In this paper, we give a complete answer for any . We show that if and is an ‐vertex ‐free graph with , then is bipartite, and the bound is tight.

The Generalized 4-Edge Connectivity of Line Graphs of the Complete Bipartite Graph

The Generalized 4-Edge Connectivity of Line Graphs of the Complete Bipartite Graph

3-Path Decompositions of the Line Graph of the Complete Bipartite Graph

The line graph is a very popular research object in graph theory, in complex networks and also in social networks recently. Let [Formula: see text] be the line graph of the complete bipartite graph [Formula: see text] and [Formula: see text] be a path of length [Formula: see text]. In this paper, we give necessary and sufficient conditions for the existence of [Formula: see text]-decompositions of [Formula: see text].

Bipartite Ramsey numbers of cycles

Let H1, …, Hk be bipartite graphs. The bipartite Ramsey number br(H1,…,Hk) is the minimum integer N such that any k-edge-coloring of the complete bipartite graph KN,N contains a monochromatic Hi in color i for some i∈{1,2,…,k}. The study of bipartite Ramsey number was initiated in the early 70s by Faudree and Schelp (1975), and they showed that br(P2n,P2m)=n+m−1, where Pn denotes a path with n vertices. Combining their result and the regularity lemma, one can obtain asymptotic value br(C2⌊α1n⌋,C2⌊α2n⌋)=(α1+α2+o(1))n for α1,α2>0, where Cn denotes a cycle with n vertices. The study of bipartite Ramsey numbers of cycles has also attracted lots of attention recently. For example, Bucić, Letzter and Sudakov (2019) showed that br(C2n,C2n,C2n)=(3+o(1))n and they remarked that it is interesting to determine the exact value of br(C2n,C2m). Zhang and Sun (2011), Zhang, Sun, and Wu (2013), and Gholami and Rowshan (2021) determined br(C4,C2n), br(C6,C2n), and br(C8,C2n) respectively. In this paper, we determine br(C2n,C2m) for all the remaining cases and show that br(C2n,C2m)=n+m−1 for all n>m≥5 and br(C2n,C2n)=2n if n≥5. It is somehow interesting that br(C2n,C2m) is the same as br(P2n,P2m) if n≠m, and br(C2n,C2n) is one more than br(P2n,P2n).

Radio number of $ 2- $ super subdivision for path related graphs

&lt;abstract&gt;&lt;p&gt;We studied radio labelings of graphs in response to the Channel Assignment Problem (CAP). In a graph $ G, $ the radio labeling is a mapping $ \varpi:V(G) \rightarrow \{0, 1, 2, ..., \}, $ such as $ |\varpi(\mu')-\varpi(\mu'')|\geq diam(G)+1-d(\mu', \mu''). $ The label of $ \mu $ for under $ \varpi $ is defined by the integer $ \varpi(\mu), $ and the span under is defined by $ span(\varpi) = max \{|\varpi(\mu')-\varpi(\mu'')|: \mu', \mu'' \in V(G)\}. $ $ rn(G) = min_{\varpi} span(\varpi) $ is defined as the radio number of $ G $ when the minimum over all radio labeling $ \varpi $ of $ G $ is taken. $ G $ is said to be optimal if its radio labeling is $ span(\varpi) = rn(G). $ A graph H is said to be an $ m $ super subdivision if $ G $ is replaced by the complete bipartite graph $ K_{m, m} $ with $ m = 2 $ in such a way that the end vertices of the edge are merged with any two vertices of the same partite set $ X $ or $ Y $ of $ K_{m, m} $ after removal of the edge of $ G $. Up to this point, many lower and upper bounds of $ rn(G) $ have been found for several kinds of graph families. This work presents a comprehensive analysis of the radio number $ rn(G) $ for a graph $ G $, with particular emphasis on the $ m $ super subdivision of a path $ P_{n} $ with $ n (n \geq 3) $ vertices, along with a complete bipartite graph $ K_{m, m} $ consisting of $ m $ v/ertices, where $ m = 2 $.&lt;/p&gt;&lt;/abstract&gt;

Graceful game on some graph classes

A graceful labeling of a graph G with m edges consists in labeling the vertices of G with distinct integers from 0 to m such that each edge is uniquely identified by the absolute difference of the labels of its endpoints. In this work, we study the graceful labeling problem in the context of maker-breaker graph games. The Graceful Game was introduced by Tuza, in 2017, as a two-players game on a connected graph in which the players, Alice and Bob, take moves labeling the vertices with distinct integers from 0 to m. Players are constrained to use only legal labelings (moves), that is, after a move, all edge labels are distinct. Alice’s goal is to obtain a graceful labeling for the graph, as Bob’s goal is to prevent it from happening. In this work, we study winning strategies for Alice and Bob in graph classes: paths, complete graphs, cycles, complete bipartite graphs, caterpillars, trees, gear graphs, web graphs, prisms, hypercubes, 2-powers of paths, wheels and fan graphs.

Newly defined fuzzy Misbalance Prodeg Index with application in multi-criteria decision-making

&lt;abstract&gt;&lt;p&gt;In crisp graph theory, there are numerous topological indices accessible, including the Misbalance Prodeg Index, which is one of the most well-known degree-based topological indexes. In crisp graphs, both vertices and edges have membership values of $ 1 $ or $ 0 $, whereas in fuzzy graphs, both vertices and edges have different memberships. This is an entire contrast to the crisp graph. In this paper, we introduce the Fuzzy Misbalance Prodeg Index of a fuzzy graph, which is a generalized form of the Misbalance Prodeg Index of a graph. We find bounds of this index and find bounds of certain classes of graphs such as path graph, cycle graph, complete graph, complete bipartite graph, and star graph. We give an algorithm to find the Fuzzy Misbalance Prodeg Index of a graph for the model of multi-criteria decision-making is established. We present applications from daily life in multi-criteria decision-making. We apply our obtained model of the Fuzzy Misbalance Prodeg Index for the multi-criteria decision-making to the choice of the best supplier and we also show the graphical analysis of our index with the other indices that show how our index is better than other existing indices.&lt;/p&gt;&lt;/abstract&gt;

The generalized k-connectivity of equally complete bipartite graphs and their line graphs

Generalized k connectivity for graphs and generic graphs is recognized as NPcomplete, a parameter that measures the network’s ability to connect vertices. Suppose km,n a completely connected bipartite graph represents the maximum number of internally disjoint Steiner trees (IDSTs) joining a subset S⸦V(G) of k vertices in G. In this context, Steiner trees (or “S -trees”) T1, T2 are considered internally detached if and only if V(T1) ∩ V(T2) = S and E(T1) ∩ E(T2) = ϕ. We define generalized k-connectivity as κk (G). This study focuses on calculating the precise values of generalized k-connectivity for line- graph of bipartite graphs with k = 3, 4 and generalized k-connectivity for bipartite graphs with k ≥ 3.

H - V -Super-Strong- ( a , d ) -antimagic decomposition of complete bipartite graphs

An H -(a,d)-antimagic labeling in a H -decomposable graph G is a bijection f : V ( G ) ∪ E ( G ) → { 1 , 2 , … , p + q } such that ∑ f ( H 1 ) , ∑ f ( H 2 ) , ⋯ , ∑ f ( H h ) forms an arithmetic progression with difference d and first element a . f is said to be H - V -super- ( a , d ) -antimagic if f ( V ( G ) ) = { 1 , 2 , … , p } . Suppose that V ( G ) = U ( G ) ∪ W ( G ) with | U ( G ) | = m and | W ( G ) | = n . Then f is said to be H - V -super-strong- ( a , d ) -antimagic labeling if f ( U ( G ) ) = { 1 , 2 , … , m } and f ( W ( G ) ) = { m + 1 , m + 2 , … , ( m + n = p ) } . A graph that admits a H - V -super-strong- ( a , d ) -antimagic labeling is called a H - V -super-strong- ( a , d ) -antimagic decomposable graph. In this paper, we prove that complete bipartite graphs K m , n are H - V -super-strong- ( a , d ) -antimagic decomposable with both m and n are even.

Triple Even Star Decomposition of Complete Bipartite Graphs

Let G be a finite, connected, undirected graph without loops or multiple edges. A decomposition {G2, G4, . . . , G2k} of G is said to be an even star decomposition if each Gi is a star and |E(Gi)| = i for all i = 2, 4, . . . , 2k. A graph G is said to have Triple Even Star Decomposition (TESD) if G can be decomposed into 3k stars {3S2, 3S4, . . . , 3S2k}. In this paper, we characterize Triple Even Star Decomposition of complete bipartite graphs&#x0D; Km,n when m = 2 and m = 3.

Complete bipartite graphs without small rainbow subgraphs

Motivated by bipartite Gallai–Ramsey type problems, we consider edge-colorings of complete bipartite graphs without rainbow tree and matching. Given two graphs G and H, and a positive integer k, define the bipartite Gallai–Ramsey numberbgrk(G:H) as the minimum number of vertices n such that n2≥k and for every N≥n, any coloring (using all k colors) of the complete bipartite graph KN,N contains a rainbow copy of G or a monochromatic copy of H. In this paper, we first describe the structures of a complete bipartite graph Kn,n without rainbow P4+ and 3K2, respectively, where P4+ is the graph consisting of a P4 with one extra edge incident with an interior vertex. Furthermore, we determine the exact values or upper and lower bounds on bgrk(G:H) when G is a 3-matching or a 4-path or P4+, and H is a bipartite graph.

Perfect state transfer by means of discrete-time quantum walk on the complete bipartite graph

Perfect state transfer has attracted a great deal of attention recently due to its crucial role in quantum communication and scalable quantum computation. In this paper, we propose the perfect state transfer algorithms with a pair of sender-receiver and two pairs of sender-receiver on the complete bipartite graph respectively. The algorithm with a pair of sender-receiver is implemented through discrete-time quantum walk, flexibly setting the coin operators based on the positions of the sender and receiver. The algorithm with two pairs of sender-receiver ensures that the two quantum states are distributed on both sides of the complete bipartite graph during the process, thereby achieving perfect state transfer. In addition, the quantum circuits corresponding to the algorithms are provided. The algorithms can transfer an arbitrary quantum state and can simultaneously transfer two arbitrary quantum states from the senders to the receivers in any case. Moreover, the algorithms are not only applicable to complete bipartite graphs but also to more graph structures with complete bipartite subgraphs, which will provide potential applications for quantum information processing.