The Maximum Spectral Radius of Theta-Free Graphs with Given Size

Spectral radius of uniform hypergraphs

Extremal graphs with bounded vertex bipartiteness number

On spectral extrema of graphs with given order and dissociation number

On the spectral radius of bipartite graphs with given diameter

The maximum spectral radius of graphs of given size with forbidden subgraph

Let $n$ and $d$ be positive integers with $1\leq d\leq n(n-1)/2$. We investigate the maximum and minimum spectral radii of a $(0,1)$-matrix of order $n$ that has $1$'s on and below its main diagonal and $d$ additional 1's. If $d\leq 4$ we determine all matrices of this type that have the maximum spectral radius. For general $d$ we prove an asymptotic result that severely limits the structure of matrices with maximum spectral radius. For $d\leq n$, we determine the minimum spectral radius.

Li, Shiu, Chan and Chang [On the spectral radius of graphs with connectivity at most k, J. Math. Chem., 46 (2009), 340-346] studied the spectral radius of graphs of order n with κ(G) ≤ k and showed that among those graphs, the maximum spectral radius is obtained uniquely at \(K_k^n\), which is the graph obtained by joining k edges from k vertices of K n − 1 to an isolated vertex. In this paper, we study the spectral radius of graphs of order n with κ(G) ≤ k and minimum degree δ(G) ≥ k . We show that among those graphs, the maximum spectral radius is obtained uniquely at K k + (K δ − k + 1 ∪ K n − δ − 1).Keywordsconnectivityspectral radius

Proof and disproof of conjectures on spectral radii of coclique extension of cycles and paths

In this paper, we first present the properties of the graph which maximize the spectral radius among all graphs with prescribed degree sequence. Using these results, we provide a somewhat simpler method to determine the unicyclic graph with maximum spectral radius among all unicyclic graphs with a given degree sequence. Moreover, we determine the bicyclic graph which has maximum spectral radius among all bicyclic graphs with a given degree sequence. Let G be a simple connected graph with vertex set V (G) and edge set E(G). Its order is |V (G)|, denoted by n, and its size is |E(G)|, denoted by m. For v ∈ V (G), let NG(v) (or N (v) for short) be the set of all neighbors of v in G and let d(v) = |N (v)| be the degree of v. We use G − e and G + e to denote the graphs obtained by deleting the edge e from G and by adding the edge e to G, respectively. For any nonempty subset W of V (G), the subgraph of G induced by W is denoted by G(W ). The distance of u and v (in G) is the length of the shortest path between u and v, denoted by d(u;v). For all other notions and definitions, not given here, see, for example, (1), or (4) (for graph spectra). For the basic notions and terminology on the spectral graph theory the readers are referred to (4). Let A(G) be the adjacency matrix of G. Its eigenvalues are called the eigenvalues

On a conjecture of spectral extremal problems

ABSTRACTIn this paper, we introduce the operations of grafting an edge and subdividing an edge on hypergraphs, and consider how spectral radius of a hypergraph behaves by grafting an edge or subdividing an edge. As an application, we determine the unique hypergraphs with the maximum spectral radius among all the uniform supertrees and all the connected uniform unicyclic hypergraphs with given number of pendant edges, respectively. Moreover, we determine the unique uniform supertree which attains the maximum spectral radius among all the uniform supertrees with given number of pendant vertices.

ABSTRACTLet G be a k-uniform hypergraph. The spectral radius of G is defined as the maximum modulus of eigenvalues of G. In this paper, we characterize the hypergraph on n vertices with maximum spectral radius among all connected k-uniform hypergraphs with exactly r pendent vertices, where or , and establish lower and upper bounds for the maximum spectral radius of connected hypergraphs in terms of the number of pendent vertices when ; We also characterize the unique hypergraphs with maximum spectral radius among all connected k-uniform hypergraphs and supertrees with given number of pendent edges, respectively.

The spectral radius of graphs with given independence number

Two spectral extremal results for graphs with given order and rank

Let dot{G}=(G,sigma ) be a signed graph, and let rho (dot{G}) (resp. lambda _1(dot{G})) denote the spectral radius (resp. the index) of the adjacency matrix A_{dot{G}}. In this paper we detect the signed graphs achieving the minimum spectral radius m(mathcal S mathcal R_n), the maximum spectral radius M(mathcal S mathcal R_n), the minimum index m(mathcal I_n) and the maximum index M(mathcal I_n) in the set mathcal U_n of all unbalanced connected signed graphs with ngeqslant 3 vertices. From the explicit computation of the four extremal values it turns out that the difference m(mathcal S mathcal R_n)-m(mathcal I_n) for n geqslant 8 strictly increases with n and tends to 1, whereas M(mathcal S mathcal R_n)- M(mathcal I_n) strictly decreases and tends to 0.