Graph Of Order Research Articles

On double domination numbers of signed complete graphs

Let [Formula: see text] be a signed graph with a vertex set [Formula: see text]. A set [Formula: see text] is said to be a double dominating set of [Formula: see text] if it satisfies the following conditions: (i) [Formula: see text] for each [Formula: see text], and (ii) [Formula: see text] is balanced, where [Formula: see text] denotes the closed neighborhood of [Formula: see text] and [Formula: see text] denotes the subgraph induced by the edges of [Formula: see text] with one end vertex in [Formula: see text] and the other end vertex in [Formula: see text]. The minimum size among all the double dominating sets of [Formula: see text] is the double domination number [Formula: see text] of [Formula: see text]. In this study, we investigated this parameter for signed complete graphs. We prove that, for [Formula: see text], if [Formula: see text] is a signed complete graph, then [Formula: see text] and these bounds are sharp. Moreover, for all signed complete graphs over [Formula: see text] we determined their possible double domination numbers. Finally, we compute the double domination numbers of all signed complete graphs of orders up to six.

Spectral Pseudorandomness and the Road to Improved Clique Number Bounds for Paley Graphs

We study subgraphs of Paley graphs of prime order p induced on the sets of vertices extending a given independent set of size a to a larger independent set. Using a sufficient condition due to Kunisky (2023), we show that estimates on a new family of necklace character sums would imply that, as p → ∞ , the empirical spectral distributions of the adjacency matrices of any sequence of such subgraphs have the same weak limit (after rescaling) as those of subgraphs induced on a random set including each vertex independently with probability 2 − a , namely, a Kesten-McKay law with parameter 2 a . We prove the necessary estimates for a = 1, obtaining in the process an alternate proof of a character sum equidistribution result of Xi (2022), and provide numerical evidence for this weak convergence for a ≥ 2 . We also conjecture that the minimum eigenvalue of any such sequence converges (after rescaling) to the left edge of the corresponding Kesten-McKay law, and provide numerical evidence for this convergence. Finally, we show that, once a ≥ 3 , this (conjectural) convergence of the minimum eigenvalue would imply bounds on the clique number of the Paley graph improving on the current state of the art due to Hanson and Petridis (2021), and that this convergence for all a ≥ 1 would imply that the clique number is o ( p ) .

The influence of separating cycles in drawings of K 5 ∖ e in the join product with paths and cycles

Abstract The crossing number cr(H) of a graph H is the minimum number of edge crossings over all drawings of H in the plane. Let H ∗ be the connected graph of order five isomorphic to K 5 ∖ e obtained by removing one edge from the complete graph K 5. The main aim of the paper is to give the crossing numbers of the join products H ∗ + Pn and H ∗ + Cn , where Pn and Cn are the path and the cycle on n vertices, respectively. The proofs are done with the help of a suitable classification of a large number of drawings of the graph H ∗ in view of the existence of a separating cycle of two possible types.

On Broadcast Schemes of Knödel Graphs

Knödel graphs on even number of vertices serve as crucial components in constructing various broadcast graphs, particularly those on an odd number of vertices. The efficacy of these constructions heavily relies on the chosen broadcast scheme. Dimensional broadcast schemes are widely employed for broadcasting Knödel graphs. This paper delves into the investigation of standard and reverse dimensional broadcast schemes identifying numerous valid schemes. Specifically, let [Formula: see text] be a Knödel graph of order [Formula: see text] and [Formula: see text]. We explore the impact of [Formula: see text] cyclic shifts on both standard and reverse dimensional broadcast schemes of Knödel graphs, where [Formula: see text], with repetition of the second dimension.

PBIB-Designs Arising From BI-Connected Dominating Sets of Cubic Graphs of Order at Most 12

A $(v, b, \gamma_{bc}, r, \lambda_{i})$-design over regular graph $G = (V, E)$ of degree $k$ is an ordered pair $D = (V, B)$, where $|V| = v$ and $B$ is the set of bi-connected dominating sets of $G$ called blocks such that two vertices $\alpha$ and $\beta$ which are $i^{th}$ associates occur together in $ \lambda_{i}$ blocks, the numbers $\lambda_{i}$ being independent of the choice of the pair $\alpha$ and $\beta$. In this paper, we obtain Partially Balanced Incomplete Block (PBIB)-designs arising from bi-connected dominating sets in cubic graphs. Also, we give a complete list of PBIB-designs with respect to the bi-connected dominating sets for cubic graphs of order at most $12$. The discussion of non-existence of some designs corresponding to bi-connected dominating sets from certain graphs concludes the article.

The λ -fold Spectrum Problem for the Orientations of the 6-Cycle

The λ-fold complete symmetric directed graph of order v, denoted λKv*, is the directed graph on v vertices and λ directed edges in each direction between each pair of vertices. For a given directed graph D, the set of all v for which λKv* admits a D-decomposition is called the λ-fold spectrum of D. In this paper, we settle the λ-fold spectrum of each of the nine non-isomorphic orientations of a 6-cycle.

Decomposition of the λ -Fold Complete Equipartite Graph into Unicyclic Graphs of Order Five

For a graph G and a subgraph H of a graph G , an H -decomposition of the graph G is a partition of the edge set of G into subsets E i , 1 ≤ i ≤ k , such that each E i induces a graph isomorphic to H . In this paper, it is proved that every simple connected unicyclic graph of order five decomposes the λ -fold complete equipartite graph whenever the necessary conditions are satisfied. This generalizes a result of Huang, *Utilitas Math.* 97 (2015), 109–117.

A Note on Graph Burning of Path Forests

Graph burning is a natural discrete graph algorithm inspired by the spread of social contagion. Despite its simplicity, some open problems remain steadfastly unsolved, notably the burning number conjecture, which says that every connected graph of order $m^2$ has burning number at most $m$. Earlier, we showed that the conjecture also holds for a path forest, which is disconnected, provided each of its paths is sufficiently long. However, finding the least sufficient length for this to hold turns out to be nontrivial. In this note, we present our initial findings and conjectures that associate the problem to some naturally impossibly burnable path forests. It is noteworthy that our problem can be reformulated as a topic concerning sumset partition of integers.

On the Signless Laplacian ABC-Spectral Properties of a Graph

In the paper, we introduce the signless Laplacian ABC-matrix Q̃(G)=D¯(G)+Ã(G), where D¯(G) is the diagonal matrix of ABC-degrees and Ã(G) is the ABC-matrix of G. The eigenvalues of the matrix Q̃(G) are the signless Laplacian ABC-eigenvalues of G. We give some basic properties of the matrix Q̃(G), which includes relating independence number and clique number with signless Laplacian ABC-eigenvalues. For bipartite graphs, we show that the signless Laplacian ABC-spectrum and the Laplacian ABC-spectrum are the same. We characterize the graphs with exactly two distinct signless Laplacian ABC-eigenvalues. Also, we consider the problem of the characterization of the graphs with exactly three distinct signless Laplacian ABC-eigenvalues and solve it for bipartite graphs and, in some cases, for non-bipartite graphs. We also introduce the concept of the trace norm of the matrix Q̃(G)−tr(Q̃(G))nI, called the signless Laplacian ABC-energy of G. We obtain some upper and lower bounds for signless Laplacian ABC-energy and characterize the extremal graphs attaining it. Further, for graphs of order at most 6, we compare the signless Laplacian energy and the ABC-energy with the signless Laplacian ABC-energy and found that the latter behaves well, as there is a single pair of graphs with the same signless Laplacian ABC-energy unlike the 26 pairs of graphs with same signless Laplacian energy and eight pairs of graphs with the same ABC-energy.

On Sombor Index of Unicyclic and Bicyclic Graphs

Gutman proposed a topological index called the Sombor index, which was defined as [Formula: see text] where [Formula: see text] is the degree of the vertex [Formula: see text] in graph [Formula: see text]. In this paper, we determine the second-minimum and second-maximum values of the Sombor index over all the unicyclic graphs of order [Formula: see text] and bicyclic graphs of order [Formula: see text].

Extremal graph of super line graph operation via generalized Randić index

Graph operations enable the modification and transformation of graphs, enhancing data representation as well as effectiveness in fields such as computer science, mathematics, and data analysis. Super line graph operations are a concept of advanced graph theory that is used in advanced mathematics and computer science. They are used to solve specific mathematical problems, especially in graph theory and Combinatorics. Topological numbers are numerical values connected with graph structures that are used to analyze structural characteristics and solve problems in fields such as chemistry and mathematics. They are important for simplifying complex data and enabling quantitative analysis. The general Randić number of some super line graph operations of number two with a three-diameter is investigated in this study. The pendant vertices are inserted without disturbing their diameter. Seven graphs of order five, each with a diameter of three, can be used to create twenty-three generalized super line graphs.

Maximum odd induced subgraph of a graph concerning its chromatic number

AbstractLet be the maximum order of an odd induced subgraph of . In 1992, Scott proposed a conjecture that for a graph of order without isolated vertices, where is the chromatic number of . In this paper, we show that the conjecture is not true for bipartite graphs, but is true for all line graphs. In addition, we also disprove a conjecture of Berman, Wang, and Wargo in 1997, which states that for a connected graph of order . Scott's conjecture is open for graphs with chromatic number at least 3.

Multi-view subspace clustering based on multi-order neighbor diffusion

Multi-view subspace clustering (MVC) intends to separate out samples via integrating the complementary information from diverse views. In MVC, since the structural information in the graph is crucial to the graph learning, most of the existing algorithms construct the superficial graph from the original data by directly measuring the similarity between the first-order complementary nearest neighbors. However, the information provided by the superficial graph structure would be influenced by contaminated or absent samples. To address this problem, in the proposed method, the higher-order complementary neighbor graphs are exploited to discover the latent structural information between the samples, and fusing the latent structural information across different orders to achieve the MVC. Specifically, the higher-order neighbor graphs under different views are leveraged to estimate the missing samples. Then, to integrate the neighbor graphs of different orders, the multi-order neighbor diffusion fusion is proposed. Nevertheless, the above problem of diffusion fusion is an intractable non-convex issue. Thus, to address it, the multi-order neighbor diffusion fusion is considered as a combination problem of the solution under different order, and the heuristic algorithm is leveraged to solve it. In this way, not only the data representation under different view and also the neighbor structure under different order can be diffused under a joint optimization framework, thus the consistency and integral information among various perspectives and orders can be utilized effectively and simultaneously. Experiments on both incomplete and complete multi-view dataset demonstrate the convincingness of the high-order neighborhood structure based subspace clustering scheme by comparing with the existing approaches.

The number of 1-nearly independent vertex subsets

Let G be a graph with vertex set V(G) and edge set E(G). A subset I of V(G) is an independent vertex subset if no two vertices in I are adjacent in G. We study the number, σ 1(G), of all subsets of V(G) that contain exactly one pair of adjacent vertices. We call those subsets 1-nearly independent vertex subsets. Recursive formulas of σ 1 are provided, as well as some cases of explicit formulas. We prove a tight lower (resp. upper) bound on σ 1 for graphs of order n. We deduce as a corollary that the star K 1,n−1 (the tree with degree sequence (n − 1, 1, . , 1)) is the n-vertex tree with smallest σ 1, while it is well known that K1,n−1 is the n-vertex tree with largest number of independent subsets.

On tricyclic graphs with maximum atom–bond sum–connectivity index

The sum-connectivity, Randić, and atom-bond connectivity indices have a prominent place among those topological indices that depend on the graph's vertex degrees. The ABS (atom-bond sum-connectivity) index is a variant of all the aforementioned three indices, which was recently put forward. Let T(n) be the class of all connected tricyclic graphs of order n. Recently, the problem of determining graphs from T(n) having the least possible value of the ABS index was solved in (Zuo et al., 2024 [39]) for the case when the maximum degree of the considered graphs does not exceed 4. The present paper addresses the problem of finding graphs from T(n) having the largest possible value of the ABS index for n≥5.

On Vertex-Disjoint Chorded Cycles and Degree Sum Conditions

In this paper, we consider a degree sum condition sufficient to imply the existence of k vertex-disjoint chorded cycles in a graph G . Let σ 4 ( G ) be the minimum degree sum of four independent vertices of G . We prove that if G is a graph of order at least 11 k + 7 and σ 4 ( G ) ≥ 12 k − 3 with k ≥ 1 , then G contains k vertex-disjoint chorded cycles. We also show that the degree sum condition on σ 4 ( G ) is sharp.

The high order spectral extremal results for graphs and their applications

The extremal problem of two types of high order spectra for graphs are considered, which are called r-adjacency spectrum and t-clique spectrum, respectively. In this paper, we obtain the maximum r-adjacency spectral radius of a Kr+1 minor-free graph of order n in the case 1≤r≤3, which implies the Hadwiger’s conjecture is true for 1≤r≤3. Moreover, an upper bound of the 3-clique spectral radius of a Bk-free and K2,l-free graph G of order n is given, where Bk is the graph consisting of k triangles sharing an edge. As a corollary of this result, we obtain an upper bound of the number of the triangles for G which improves a result of Alon and Shikhelman (2016).

Distance magic labelings of Cartesian products of cycles

A graph of order n is distance magic if it admits a bijective labeling of its vertices with integers from 1 to n such that each vertex has the same sum of the labels of its neighbors. In this paper we classify all distance magic Cartesian products of two cycles, thereby correcting an error in a widely cited paper from 2004. Additionally, we show that each distance magic labeling of a Cartesian product of cycles is determined by a pair or quadruple of suitable sequences, thus obtaining a complete characterization of all distance magic labelings of these graphs. We also determine a lower bound on the number of all distance magic labelings of Cm□C2m with m≥3 odd.

A tight upper bound on the average order of dominating sets of a graph

AbstractIn this paper we study the average order of dominating sets in a graph, . Like other average graph parameters, the extremal graphs are of interest. Beaton and Brown conjectured that for all graphs of order without isolated vertices, . Recently, Erey proved the conjecture for forests without isolated vertices. In this paper we prove the conjecture and classify which graphs have . We also use our bounds to prove an average version of Vizing's conjecture.

Proof of a conjecture on dominating sets inducing large component in graphs with minimum degree two

The k-component domination number γk(G) of G is the minimum cardinality of a dominating set S of G such that each component of G[S] has order at least k. Clearly, γ1(G) is the domination number of G and γ2(G) is the total domination number of G. With a finite set of exceptional graphs, McCuaig and Shepherd (1989) [13] proved that γ1(G)≤25n and Henning (2000) [11] proved that γ2(G)≤47n respectively. As an extension, Alvarado et al. (2016) [1] posed a conjecture which says that if G is a connected graph of order n≥k+1 and δ(G)≥2, then γk(G)≤2kn2k+3 unless G belongs to a set of exceptional graphs. In this paper, we confirm the validity of the above conjecture. Moreover, we characterize those graphs of order n which are edge-minimal with respect to satisfying G connected, δ(G)≥2 and γk(G)≥2kn2k+3. This strengthens a result of Alvarado, Dantas, and Rautenbach in the same paper.