Extremal Graphs Research Articles

On the number of Fk,4-saturating edges

For a graph F, let G be an F-free graph, a non-edge e of G is an F-saturating edge if G+e contains a copy of F. Graph Fk,r consists of k cliques Kr intersecting in exactly one common vertex. Denote by fF(n,m) the minimum number of F-saturating edges of F-free graphs on n vertices with m edges and fF⁎(n,m), where m≤ex(n,F), the minimum number of F-saturating edges of F-free graphs on n vertices with m edges obtained by deleting edges from the extremal graph attaining ex(n,F). In this paper, we study the number of Fk,4-saturating edges in Fk,4-free graphs on n vertices with ex(n,Fk−1,4)+1 edges. We give the upper bounds on fFk,4(n,ex(n,Fk−1,4)+1) and get the value of fF2,4(n,ex(n,F1,4)+1). Moreover, we characterize the extremal graphs attaining ex(n,Fk,4) with odd k≥3 and prove fFk,4⁎(n,ex(n,Fk−1,4)+1)=⌊n3⌋−k for odd k≥3.

Principal eigenvectors in hypergraph Turán problems

For a general class of hypergraph Turán problems with uniformity $r$, we investigate the principal eigenvector for the $p$-spectral radius (in the sense of Keevash-Lenz-Mubayi and Nikiforov) for the extremal graphs, showing in a strong sense that these eigenvectors have close to equal weight on each vertex (equivalently, showing that the principal ratio is close to $1$). We investigate the sharpness of our result; it is likely sharp for the Turán tetrahedron problem. In the course of this latter discussion, we establish a lower bound on the $p$-spectral radius of an arbitrary $r$-graph in terms of the degrees of the graph. This builds on earlier work of Cardoso-Trevisan, Li-Zhou-Bu, Cioabă-Gregory, and Zhang. The case $1 < p < r$ of our results leads to some subtleties connected to Nikiforov's notion of $k$-tightness, arising from the Perron-Frobenius theory for the $p$-spectral radius. We raise a conjecture about these issues and provide some preliminary evidence for our conjecture.

Spectral Extremal Problem on $t$ Copies of $\ell$-Cycles

Extremal problem on cycles plays an important role in extremal graph theory. Let $ex(n,F)$ and $spex(n,F)$ be the maximum size and spectral radius over all $n$-vertex $F$-free graphs, respectively. In this paper, we shall pay attention to the study of both $ex(n,tC_\ell)$ and $spex(n,tC_\ell)$. On the one hand, we determine $ex(n,tC_{2\ell+1})$ and characterize the extremal graph for any integers $t,\ell$ and $n\geq f(t,\ell)$, where $f(t,\ell)=O(t\ell^2)$. This generalizes the result on $ex(n,tC_3)$ of Erdős [Arch. Math. 13 (1962) 222–227] as well as the research on $ex(n,C_{2\ell+1})$ of Füredi and Gunderson [Combin. Probab. Comput. 24 (2015) 641–645]. On the other hand, motivated by the spectral Turán-type problem proposed by Nikiforov, we obtain the extremal spectral radius $spex(n,tC_{\ell})$ for any fixed $t,\ell$ and large enough $n$. Our results extend some classic spectral extremal results or conjectures on odd cycles and even cycles. Our results also give some inspirations for general spectral Turán-type problem $spex(n,F)$ on bipartite or non-partite $F$.

A localized approach for Turán number of long cycles

AbstractLet be a simple graph and denote the length of the longest cycle containing an edge if there exists a cycle containing , and 2 otherwise. We prove that the summation of 2 divided by over all edges in an ‐vertex graph is at most , and characterize all ‐vertex extremal graphs that achieve the bound, thereby extending the classic Erdős–Gallai theorem on cycles.

The Multiplicative Sum Zagreb Indices of Graphs with Given Clique Number

The multiplicative sum Zagreb index is a modified version of the well-known Zagreb indices. The multiplicative sum Zagreb index of a graph G is the product of the sums of the degrees of pairs of adjacent vertices. The mathematical properties of the multiplicative sum Zagreb index of graphs with given graph parameters deserve further study, as they can be used to detect chemical compounds and study network structures in mathematical chemistry. Therefore, in this paper, the maximal and minimal values of the multiplicative sum Zagreb indices of graphs with a given clique number are presented. Furthermore, the corresponding extremal graphs are characterized.

Minimizing the Laplacian-energy-like of graphs

Let G be a connected simple graph with order n and Laplacian matrix L(G). The Laplacian-energy-like of G is defined asLEL(G)=∑i=1nλi, where λi is the eigenvalue of L(G) for i=1,…,n. In this paper, with the aid of Ferrers diagrams of threshold graphs, we provide an algebraic combinatorial approach to determine the graphs with minimal Laplacian-energy-like among all connected graphs having n vertices and m edges, showing that the extremal graph is a special threshold graph, named as the quasi-complete graph.

On spectral extrema of graphs with given order and generalized 4-independence number

Characterizing the graph having the maximum or minimum spectral radius in a given class of graphs is a classical problem in spectral extremal graph theory, originally proposed by Brualdi and Solheid. Given a graph G, a vertex subset S is called a maximum generalized 4-independent set of G if the induced subgraph G[S] dose not contain a 4-tree as its subgraph, and the subset S has maximum cardinality. The cardinality of a maximum generalized 4-independent set is called the generalized 4-independence number of G. In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest spectral radius over all connected graphs (resp. bipartite graphs, trees) with fixed order and generalized 4-independence number, in addition, we establish a lower bound on the generalized 4-independence number of a tree with fixed order. Secondly, we describe the structure of all the n-vertex graphs having the minimum spectral radius with generalized 4-independence number ψ, where ψ⩾⌈3n/4⌉. Finally, we identify all the connected n-vertex graphs with generalized 4-independence number ψ∈{3,⌈3n/4⌉,n−1,n−2} having the minimum spectral radius.

Extremal graphs without exponentially small bicliques

Extremal graphs without exponentially small bicliques

On the Extremal Weighted Mostar Index of Bicyclic Graphs

Let G be a simple connected graph with edge set E(G) and vertex set V(G). The weighted Mostar index of a graph G is defined as w+Mo(G)=∑e=uv∈E(G)(dG(u)+dG(v))|nu(e)−nv(e)|, where nu(e) denotes the number of vertices closer to u than to v for an edge uv in G. In this paper, we obtain the upper bound and lower bound of the weighted Mostar index among all bicyclic graphs and characterize the corresponding extremal graphs.

Maximal diameter of integral circulant graphs

Integral circulant graphs are proposed as models for quantum spin networks enabling perfect state transfer. Understanding the potential information transfer between nodes in such networks involves calculating the maximal graph diameter. The integral circulant graph ICGn(D) has vertex set Zn={0,1,2,…,n−1}, with vertices a and b adjacent if gcd⁡(a−b,n)∈D, where D⊆{d:d|n,1≤d<n}. Building on the upper bound 2|D|+1 for the diameter provided by Saxena, Severini, and Shparlinski, we prove that the maximal diameter of ICGn(D) for a given order n with prime factorization p1α1⋯pkαk is r(n) or r(n)+1, where r(n)=k+|{i|αi>1,1≤i≤k}|. We show that a divisor set D with |D|≤k achieves this bound. We calculate the maximal diameter for graphs of order n and divisor set cardinality t≤k, identifying all extremal graphs and improving the previous upper bound.

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.

Extremal graphs with given parameters in respect of general ABS index

For a graph G=(V,E), the general atom-bond sum-connectivity index is formulated by ABSσ(G)=∑ξζ∈E(G)(dξ+dζ−2dξ+dζ)σ, where dξ indicates the degree of vertex ξ∈V, σ can be arbitrary real number. Several physicochemical features of benzenoid hydrocarbons can be correctly anticipated using the ABSσ index. Researchers go through careful inspection of some ABSσ and reveal that ABS1, ABS−1, ABS12, ABS3 enjoy benefit in foreseeing the boiling point, the standard enthalpy of vaporization, the acentric factor, and the entropy of octane isomers, separately. One significant wellspring of information for exploring the molecular construction is the assessed worth of the reach for topological indices through certain predefined molecular graph parameters. The purpose of the present work is to find the highest values of the ABSσ index for σ≥0 in the set of graphs that were given certain predefined parameters, such as matching number, chromatic number, etc. Furthermore, illustrations of the relevant extremal graphs were provided.

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.

On the neighbourhood degree sum-based Laplacian energy of graphs

A useful extension of the Laplacian matrix is proposed here and the corresponding modification of the Laplacian energy (LE) is presented. The neighbourhood degree sum-based Laplacian energy (LNE) is produced by means of the eigenvalues of the newly introduced neighbourhood degree sum-based Laplacian matrix (LN). We investigate the mathematical properties of LNE by comparing it with the Laplacian energy. The role of LNE in structure–property modelling of molecules is demonstrated by statistical approach. The formulation of LNE is not ad hoc; rather, its chemical significance exerts that it outperforms LE in modelling physiochemical behaviours of molecules. Mathematical properties of LN are also revealed by finding crucial bounds of its eigenvalues with characterizing extremal graphs.

Extremal graphs with respect to modified Sombor index

The modified Sombor (mSO) index of a given graph [Formula: see text] is symbolized with mSO and define by sum of the weights [Formula: see text], where [Formula: see text] is the degree of a vertex [Formula: see text] in [Formula: see text]. In this work, we present the first and second maximum values of the mSO index among all [Formula: see text]-vertex [Formula: see text]-cyclic graphs. We also provide that the mSO index attains its maximum and second maximum on two classes of chemical graphs. Finally, we will present new lower and upper bounds for the mSO index of connected chemical graphs.

Some results involving the A α -eigenvalues for graphs and line graphs

Abstract Let G G be a simple graph with adjacency matrix A ( G ) A\left(G) , degree diagonal matrix D ( G ) , D\left(G), and let l ( G ) l\left(G) be the line graph of G G . In 2017, Nikiforov defined the A α {A}_{\alpha } -matrix of G G , A α ( G ) {A}_{\alpha }\left(G) , as a linear convex combination of A ( G ) A\left(G) and D ( G ) D\left(G) , in the following way, A α ( G ) ≔ α A ( G ) + ( 1 − α ) D ( G ) , {A}_{\alpha }\left(G):= \alpha A\left(G)+\left(1-\alpha )D\left(G), where α ∈ [ 0 , 1 ] \alpha \in \left[0,1] . In this study, we present some bounds for the largest eigenvalue of A α ( G ) , {A}_{\alpha }\left(G), and for some eigenvalues of A α ( l ( G ) ) {A}_{\alpha }\left(l\left(G)) . Extremal graphs attaining some of these bounds are also characterized. Furthermore, some comparisons between the new bounds obtained in this study, and between these and some bounds presented by Nikiforov are made.

Extremal properties of connected 4-cyclic graphs of a topological index related to chemical graphs.

Augmented Zagreb index (AZI) is an important vertex-degree-based topological index with many applications especially in chemistry. In this article, minimum value of is obtained and the corresponding extremal graph is characterized in the class of connected 4-cyclic graphs with seven or more vertices. These results can be used to characterize numerous chemical properties of chemical compounds having 4-cyclic structures.

Spectral properties of Sombor matrix of threshold graphs

We investigate the Sombor spectral properties of threshold graphs, a formula for the Sombor index is presented, the Sombor eigenvalues are given, graphs with simple Sombor eigenvalues are characterized, bounds on the smallest/largest Sombor eigenvalues are presented, the multiplicities of the Sombor eigenvalues are discussed, formulae for trace and determinant of the associated quotient matrix are given, the Sombor spread bound and the bounds on the Sombor energy along with the characterization of extremal graphs. At the end, the conclusion states that all our results are valid for adjacency matrix and other adjacency type matrices.

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.

[formula omitted]-isolation in graphs

Let G be a graph and H be a connected graph. A subset D⊆V(G) is called an H-isolating set of G if G−N[D] contains no H as a subgraph. The H-isolation number of G, denoted by ι(G,H), is the minimum cardinality of an H-isolating set in G. For an integer k≥0, a K1,k+1-isolating set of G is simply said to be a k-isolating set of G, and ι(G,K1,k+1) is abbreviated to ιk(G), called the k-isolation number of G. Thus, ιk(G) is the minimum cardinality of a k-isolating set D such that Δ(G−N[D])≤k. We prove that if G is a connected graph of size m, then ιk(G)≤m+1k+3, unless G≅K1,k+1, or k=1 and G≅C6. The extremal graphs are completely characterized. Moreover, it is proved that for any graph G with s components, ι(G,H)≤γ(H)m(G)+sm(H)+1. Several conjectures are proposed in the closing, where γ(H) is the domination number of H.