Maximum Spectral Radius Research Articles

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.

A stability result for Berge-[formula omitted][formula omitted]-graphs and its applications

An r-uniform hypergraph (r-graph) is linear if any two edges intersect at most one vertex. For a graph F, a hypergraph H is a Berge-F if there is a bijection ϕ:E(F)→E(H) such that e⊆ϕ(e) for all e in E(F). In this paper, a kind of stability result for Berge-K3,t linear r-graphs is established. Based on this stability result, an upper bound for the linear Turán number of Berge-K3,t is determined. For an r-graph H, let A(H) be the adjacency tensor of H. The spectral radius of H is the spectral radius of the tensor A(H). Some bounds for the maximum spectral radius of connected Berge-K3,t-free linear r-graphs are obtained.

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).

Spectral extremal graphs for edge blow-up of star forests

The edge blow-up of a graph G, denoted by Gp+1, is obtained by replacing each edge of G with a clique of order p+1, where the new vertices of the cliques are all distinct. Yuan (2022) [20] determined the range of the Turán numbers for edge blow-up of all bipartite graphs and the exact Turán numbers for edge blow-up of all non-bipartite graphs. In this paper, we prove that the graphs with the maximum spectral radius in an n-vertex graph without any copy of edge blow-up of star forests are the extremal graphs for edge blow-up of star forests when n is sufficiently large.

Spectral Extremal Results on Trees

Let ${\rm spex}(n,F)$ be the maximum spectral radius over all $F$-free graphs of order $n$, and ${\rm SPEX}(n,F)$ be the family of $F$-free graphs of order $n$ with spectral radius equal to ${\rm spex}(n,F)$. Given integers $n,k,p$ with $n>k>0$ and $0\leq p\leq \lfloor(n-k)/2\rfloor$, let $S_{n,k}^{p}$ be the graph obtained from $K_k\nabla(n-k)K_1$ by embedding $p$ independent edges within its independent set, where '$\nabla$' means the join product. For $n\geq\ell\geq 4$, let $G_{n,\ell}=S_{n,(\ell-2)/2}^{0}$ if $\ell$ is even, and $G_{n,\ell}=S_{n,(\ell-3)/2}^{1}$ if $\ell$ is odd. Cioabă, Desai and Tait [SIAM J. Discrete Math. 37 (3) (2023) 2228-2239] showed that for $\ell\geq 6$ and sufficiently large $n$, if $\rho(G)\geq \rho(G_{n,\ell})$, then $G$ contains all trees of order $\ell$ unless $G=G_{n,\ell}$. They further posed a problem to study ${\rm spex}(n,F)$ for various specific trees $F$. Fix a tree $F$ of order $\ell\geq 6$, let $A$ and $B$ be two partite sets of $F$ with $|A|\leq |B|$, and set $q=|A|-1$. We first show that any graph in ${\rm SPEX}(n,F)$ contains a spanning subgraph $K_{q,n-q}$ for $q\geq 1$ and sufficiently large $n$. Consequently, $\rho(K_{q,n-q})\leq {\rm spex}(n,F)\leq \rho(G_{n,\ell})$, we further respectively characterize all trees $F$ with these two equalities holding. Secondly, we characterize the spectral extremal graphs for some specific trees and provide asymptotic spectral extremal values of the remaining trees. In particular, we characterize the spectral extremal graphs for all spiders, surprisingly, the extremal graphs are not always the spanning subgraph of $G_{n,\ell}$.

Spectral extrema of 1-planar graphs

A graph G is called a 1-planar graph if G has a drawing in the plane, so that each of its edges is crossed at most once. In this paper, we prove that λ1(G)≤5+2n+5 for a 1-planar graph G of order n≥7. For n≥6.23×1018, we characterize the unique extremal graph with the maximum spectral radius among all 1-planar graphs of order n.

The Graft Transformation and Their Application on the Spectral Radius of Block Graphs

Let [Formula: see text] be a connected graph and [Formula: see text] be the adjacency matrix of [Formula: see text]. Suppose that [Formula: see text] are the eigenvalues of [Formula: see text]. In this paper, we first give a graft transformation on the spectral radius of graphs and then as their application, we determine the extremal graphs with maximum and minimum spectral radii among all clique trees. Furthermore, we also determine the unique graph with maximum spectral radius among all block graphs by using different methods.

Extremal values for the spectral radius of the normalized distance Laplacian

The normalized distance Laplacian of a graph G is defined as DL(G)=T(G)−1/2(T(G)−D(G))T(G)−1/2 where D(G) is the matrix with pairwise distances between vertices and T(G) is the diagonal transmission matrix. In this project, we study the minimum and maximum spectral radii associated with this matrix, and the structures of the graphs that achieve these values. In particular, we prove a conjecture of Reinhart that the complete graph is the unique graph with minimum spectral radius, and we give several partial results towards a second conjecture of Reinhart regarding which graph has the maximum spectral radius.

Spectral Turán problem for [formula omitted]-free signed graphs

The classical spectral Turán problem is to determine the maximum spectral radius of an F-free graph of order n. Let Kk− be the set of all unbalanced Kk. Wang, Hou and Li [25] and Chen and Yuan [10] studied the existence of unbalanced K3 and K4, respectively. In this paper, we focus on the spectral Turán problem of Kk−-free signed graph for k≥5. By using a different method, we give an answer for k=5 and completely characterize the corresponding extremal signed graph.

Spectral extremal problem on the square of ℓ-cycle

Let Cℓ be the cycle of order ℓ. The square of Cℓ, denoted by Cℓ2, is obtained by joining all pairs of vertices with distance no more than two in Cℓ. Denote by ex(n,F) and spex(n,F) the maximum size and maximum spectral radius over all n-vertex F-free graphs, respectively. The well-known Turán problem asks for ex(n,F), and Nikiforov in 2010 proposed a spectral counterpart, known as the Brualdi-Solheid-Turán type problem, focusing on determining spex(n,F). In this paper, for any integer ℓ≥6 that is not divisible by 3, we characterize the unique extremal graph with respect to ex(n,Cℓ2) and spex(n,Cℓ2) for sufficiently large n, respectively.

Independence number and spectral radius of cactus graphs

Four Color Theorem implies that the independence number of every planar graph is at least n/4. A cactus graph is a connected planar graph whose block is either an edge or a cycle. In this paper, the lower bound for independence number of cactus graphs is improved to ⌈n/3⌉. Moreover, the maximum spectral radius of cactus graphs with fixed independence number is determined.

Extremal spectral radius and essential edge-connectivity

An edge-cut X of G is essential if G−X has at least two non-trivial components, the essential edge-connectivity of G is the minimum cardinality over all essential edge-cuts of G. In this paper, we determine the graphs attaining the maximum spectral radius among all graphs with minimum degree and essential edge-connectivity. Moreover, the extremal graphs are characterized. This generalizes some previous results on edge-connectivity.

Extremal spectral results of planar graphs without vertex‐disjoint cycles

AbstractGiven a planar graph family , let and be the maximum size and maximum spectral radius over all ‐vertex ‐free planar graphs, respectively. Let be the disjoint union of copies of ‐cycles, and be the family of vertex‐disjoint cycles without length restriction. Tait and Tobin determined that is the extremal spectral graph among all planar graphs with sufficiently large order , which implies the extremal graphs of both and for are . In this paper, we first determine and and characterize the unique extremal graph for , and sufficiently large . Second, we obtain the exact values of and , which solve a conjecture of Li for .

Alon–Boppana-Type Bounds for Weighted Graphs

The unraveled ball of radius $r$ centered at a vertex $v$ in a weighted graph $G$ is the ball of radius $r$ centered at $v$ in the universal cover of $G$. We present a general bound on the maximum spectral radius of unraveled balls of fixed radius in a weighted graph.
 The weighted degree of a vertex in a weighted graph is the sum of weights of edges incident to the vertex. A weighted graph is called regular if the weighted degrees of its vertices are the same. Using the result on unraveled balls, we prove a variation of the Alon–Boppana theorem for regular weighted graphs.

Maximum degree and spectral radius of graphs in terms of size

Maximum degree and spectral radius of graphs in terms of size

Global stability and optimal vaccination control of SVIR models

<abstract> <p>Vaccination is widely acknowledged as an affordable and cost-effective approach to guard against infectious diseases. It is important to take vaccination rate, vaccine effectiveness, and vaccine-induced immune decline into account in epidemic dynamical modeling. In this paper, an epidemic dynamical model of vaccination is developed. This model provides a framework of the infectious disease transmission dynamics model through qualitative and quantitative analysis. The result shows that the system may have multiple equilibria. We used the next-generation operator approach to calculate the maximum spectral radius, that is, basic reproduction number $ {R_{vac}} $. Next, by dividing the model into infected and uninfected subjects, we can prove that the disease-free equilibrium is globally asymptotically stable when $ {R_{vac}} < 1 $, provided certain assumptions are satisfied. When $ {R_{vac}} > 1 $, there exists a unique endemic equilibrium. Using geometric methods, we calculate the second compound matrix and demonstrate the Lozinskii measure $ \bar q \leqslant 0 $, which is equivalent to the unique endemic equilibrium, which is globally asymptotically stable. Then, using center manifold theory, we justify the existence of forward bifurcation. As the vaccination rate decreases, the likelihood of forward bifurcation increases. We also theoretically show the presence of Hopf bifurcation. Then, we performed sensitivity analysis and found that increasing the vaccine effectiveness rate can curb the propagation of disease effectively. To examine the influence of vaccination on disease control, we chose the vaccination rate as the optimal vaccination control parameter, using the Pontryagin maximum principle, and we found that increasing vaccination rates reduces the number of infected individuals. Finally, we ran a numerical simulation to finalize the theoretical results.</p> </abstract>

Spectral Turán problems for intersecting even cycles

Let C2k1,2k2,…,2kt denote the graph obtained by intersecting t distinct even cycles C2k1,C2k2,…,C2kt at a unique vertex. In this paper, we determine the unique graph with maximum adjacency spectral radius among all graphs on n vertices that do not contain any C2k1,2k2,…,2kt as a subgraph, for n sufficiently large. When one of the constituent even cycles is a C4, our results improve upper bounds on the Turán numbers for intersecting even cycles that follow from more general results of Füredi [21] and Alon, Krivelevich and Sudakov [1]. Our results may be seen as extensions of previous results for spectral Turán problems on forbidden even cycles C2k,k≥2 (see [8,36,46,47]).

Spectral extrema of [formula omitted]-free graphs

For a set of graphs F, a graph is said to be F-free if it does not contain any graph in F as a subgraph. Let Exsp(n,F) denote the graphs with the maximum spectral radius among all F-free graphs of order n. A linear forest is a graph whose connected components are paths. Denote by Ls the family of all linear forests with s edges. In this paper the graphs in Exsp(n,{Kk+1,Ls}) will be completely characterized when n is appropriately large.

On spectral extrema of graphs with given order and dissociation number

Identifying the graph with maximum or minimum spectral radius among a given class of graphs is a central problem in extremal spectral graph theory, known as the Brualdi–Solheid problem. For a graph G=(VG,EG), a subset S⊆VG is called a maximum dissociation set if the induced subgraph G[S] does not contain P3 as its subgraph, and the subset has maximum cardinality. The dissociation number of G is the number of vertices in a maximum dissociation set of G. In this paper, we first characterize all the connected graphs (resp. bipartite graphs, trees) having maximum spectral radius among connected graphs (resp. bipartite graphs, trees) with given order and dissociation number. Secondly, we show that the connected n-vertex graph with dissociation number φ having minimum spectral radius is a tree, where φ≥23n. Finally, we determine the graphs having minimum spectral radius with fixed order n and dissociation number φ∈{2,⌈2n/3⌉,n−1,n−2}.

Spectral extremal graphs for the bowtie

Let Fk be the (friendship) graph obtained from k triangles by sharing a common vertex. The Fk-free graphs of order n which attain the maximal spectral radius was firstly characterized by Cioabă, Feng, Tait and Zhang [Electron. J. Combin. 27 (4) (2020)], and later uniquely determined by Zhai, Liu and Xue [Electron. J. Combin. 29 (3) (2022)] under the condition that n is sufficiently large. In this paper, we get rid of the condition on n being sufficiently large if k=2. The graph F2 is also known as the bowtie. We show that the unique n-vertex F2-free spectral extremal graph is the balanced complete bipartite graph adding an edge in the vertex part with smaller size if n≥7, and the condition n≥7 is tight. Our result is a spectral generalization of a theorem of Erdős, Füredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995)], which states that ex(n,F2)=⌊n2/4⌋+1. Moreover, we study the spectral extremal problem for Fk-free graphs with given number of edges. In particular, we show that the unique m-edge F2-free spectral extremal graph is the join of K2 with an independent set of m−12 vertices if m≥8, and the condition m≥8 is tight.