For a graph $H$, a graph $G$ is $H$-saturated if $G$ does not contain $H$ as a subgraph but for any $e \in E(\overline{G})$, $G+e$ contains $H$. In this note, we prove a sharp lower bound for the number of paths and walks on length $2$ in $n$-vertex $K_{r+1}$-saturated graphs. We then use this bound to give a lower bound on the spectral radii of such graphs which is asymptotically tight for each fixed $r$ and $n\to\infty$.
Write $ρ\left( G\right) $ for the spectral radius of a graph $G$ and $S_{n,r}$ for the join $K_{r}\vee\overline{K}_{n-r}.$ Let $n>r\geq2$ and $G$ be a $K_{r+1}$-saturated graph of order $n.$ Recently Kim, Kim, Kostochka, and O determined exactly the minimum value of $ρ\left( G\right) $ for $r=2$, and found an asymptotically tight bound on $ρ\left( G\right) $ for $r\geq3.$ They also conjectured that \[ ρ\left( G\right) >ρ\left( S_{n,r-1}\right) , \] unless $G=S_{n,r-1}.$ In this note their conjecture is proved.
The $α$-spectral radius of a connected graph $G$ is the spectral radius of $A_α$-matrix of $G$. In this paper, we discuss the methods for comparing $α$-spectral radius of graphs. As applications, we characterize the graphs with the maximal $α$-spectral radius among all unicyclic and bicyclic graphs of order $n$ with diameter $d$, respectively. Finally, we determine the unique graph with maximal signless Laplacian spectral radius among bicyclic graphs of order $n$ with diameter $d$.
The generalized distance spectral radius of a connected graph $G$ is the spectral radius of the generalized distance matrix of $G$, defined by $$D_α(G)=αTr(G)+(1-α)D(G), \;\;0\leα\le 1,$$ where $D(G)$ and $Tr(G)$ denote the distance matrix and diagonal matrix of the vertex transmissions of $G$, respectively. This paper characterizes the unique graph with minimum generalized distance spectral radius among the connected graphs with fixed chromatic number, which answers a question about the generalized distance spectral radius in spectral extremal theories. In addition, we also determine graphs w…
Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$. The saturation number and extremal number of $\mathcal{F}$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in\mathbb{N}$, let $\mathcal{F}_k$ and $\mathcal{F}'_k$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n…
The spectral radius (or the signless Laplacian spectral radius) of a general hypergraph is the maximum modulus of the eigenvalues of its adjacency (or its signless Laplacian) tensor. In this paper, we firstly obtain a lower bound of the spectral radius (or the signless Laplacian spectral radius) of general hypergraphs in terms of clique number. Moreover, we present a relation between a homogeneous polynomial and the clique number of general hypergraphs. As an application, we finally obtain an upper bound of the spectral radius of general hypergraphs in terms of clique number.
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative weakly irreducible tensor. We also apply these bounds to the adjacency spectral radius and signless Laplacian spectral radius of a uniform hypergraph.
For $0\leα<1$ and a uniform hypergraph $G$, the $α$-spectral radius of $G$ is the largest $H$-eigenvalue of $α\mathcal{D}(G) +(1-α)\mathcal{A}(G)$, where $\mathcal{D}(G)$ and $\mathcal{A}(G)$ are the diagonal tensor of degrees and the adjacency tensor of $G$, respectively. We give upper bounds for the $α$-spectral radius of a uniform hypergraph, propose some transformations that increase the $α$-spectral radius, and determine the unique hypergraphs with maximum $α$-spectral radius in some classes of uniform hypergraphs.
In this paper we show how the $α$-spectral radius changes under the edge grafting operations on connected $k$-uniform hypergraphs. We characterize the extremal hypertree for $α$-spectral radius among $k$-uniform non-caterpillar hypergraphs with given order, size and diameter. we also characerize the second largest $α$-spectral radius among all $k$-uniform supertrees on $n$ vertices by two methods.
The spectral radius of a uniform hypergraph is defined to be that of the adjacency tensor the hypergraph. It is known that the unique unicyclic hypergraph with the largest spectral radius is a nonlinear hypergraph, and the unique linear unicyclic hypergraph with the largest spectral radius is a power hypergraph. In this paper we determine the the unique linear unicyclic hypergraph with the second or third largest spectral radius, where the former hypergraph is a power hypergraph and the latter hypergraph is a non-power hypergraph.
Let $G$ be a graph of order $n$ and spectral radius be the largest eigenvalue of its adjacency matrix, denoted by $μ(G)$. In this paper, we determine the unique graph with maximum spectral radius among all graphs of order $n$ without containing the $2$-power of a Hamilton cycle.
A cactus is a connected graph in which any two cycles have at most one common vertex. We determine the unique graph that maximizes the distance spectral radius over all cacti with fixed numbers of vertices and cycles, and thus prove a conjecture on the distance spectral radius of cacti in [S.S. Bose, M. Nath, S. Paul, On the distance spectral radius of cacti, Linear Algebra Appl. 437 (2012) 2128--2141]. We prove the result in the context of hypertrees.
For a nonnegative weakly irreducible tensor, its spectral radius is an eigenvalue corresponding to a unique positive eigenvector up to a scalar called the Perron vector. But including the Perron vector, it may have more than one eigenvector corresponding to the spectral radius. The projective eigenvariety associated with the spectral radius is the set of the eigenvectors corresponding to the spectral radius considered in the complex projective space. In this paper we prove that the dimension of the above projective eigenvariety is zero, i.e. there are finite many eigenvectors associated with t…
Let $\mathscr{G}_{n,β}$ be the set of graphs of order $n$ with given matching number $β$. Let $D(G)$ be the diagonal matrix of the degrees of the graph $G$ and $A(G)$ be the adjacency matrix of the graph $G$. The largest eigenvalue of the nonnegative matrix $A_α(G)=αD(G)+A(G)$ is called the $α$-spectral radius of $G$. The graphs with maximal $α$-spectral radius in $\mathscr{G}_{n,β}$ are completely characterized in this paper. In this way we provide a general framework to attack the problem of extremal spectral radius in $\mathscr{G}_{n,β}$. More precisely, we generalize the known results on t…
In this paper, using the theory of matching polynomial of hypertrees and ordering of hypertrees, we determine the largest spectral radius of hypertrees with $m$ edges and given size of matching.
Let $\mathcal{H}^{(r)}_n$ be the set of all connected $r$-graphs with given size $n$. In this paper, we investigate the effect on the spectral radius of $r$-uniform hypergraphs by grafting or contracting an edge and then give the ordering of the $r$-graphs with small spectral radius over $\mathcal{H}^{(r)}_n$, when $n\geq 20$.
We show that the kite graph $K_4^{(n)}$ uniquely maximizes the distance spectral radius among all connected $4$-chromatic planar graphs on $n$ vertices.
Let $λ^{*}$ be the maximum spectral radius of connected irregular graphs on $n$ vertices with maximum degree $Δ$. Liu, Shen and Wang (2007) conjectured that $\lim_{n\rightarrow \infty}(n^{2}(Δ-λ^{*}))/(Δ-1)=π^{2},$ which describes the asymptotic behavior for the maximum spectral radius of irregular graphs. Focusing on this conjecture, we consider the maximum spectral radius of connected subcubic bipartite graphs. The unique connected subcubic bipartite graph with the maximum spectral radius is determined. Let $G$ be a $k$-connected irregular graph with spectral radius $λ_{1}(G)$, we present a …
In this paper, we study the Hamiltonicity of graphs with large minimum degree. Firstly, we present some conditions for a simple graph to be Hamilton-connected and traceable from every vertex in terms of the spectral radius of the graph or its complement respectively. Secondly, we give the conditions for a nearly balanced bipartite graph to be traceable in terms of spectral radius, signless Laplacian spectral radius of the graph or its quasi-complement respectively.
Let $\mathcal{G}_{n, β^*}$ $(\mathcal{G}^*_{n,β^*})$ be the set of all (connected) graphs of order $n$ with fractional matching number $β^*$. In this paper, the graphs with maximal spectral radius in $\mathcal{G}_{n,β^*}$ and $\mathcal{G}^*_{n,β^*}$ are characterized, respectively. Moreover, a lower bound for the spectral radius in graphs with order $n$ to guarantee the existence of a perfect fractional matching is also given, which generalizes the main result of O [Suil O, Spectral radius and matchings in graphs, Linear Algebra and its Applications, 2020].
Some sharp bounds on the distance signless Laplacian spectral radius of graphs
M. Aouchiche和P. Hansen提出了连通图的距离Laplacian和距离无符号Laplacian [Two Laplacian for the distance matrix of a graph，LAA 439（2013）21{33]。本文给出了非负矩阵谱半径的三个精确上界定理，并将这些定理分别应用于无符号Laplacian矩阵和距离无符号Laplacian矩阵，得到了谱半径的精确上界.我们还提出了一个关于无符号Laplacian谱半径的锐界的已知结果存在缺陷。
For $0\le α\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_α(G)=αD(G)+(1-α)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix. The $α$-spectral radius of $G$ is the largest eigenvalue of $A_α(G)$. We give upper bounds for $α$-spectral radius for unicyclic graphs $G$ with maximum degree $Δ\ge 2$, connected irregular graphs with given maximum degree and and some other graph parameters, and graphs with given domination number, respectively. We determine the unique tree with second maximum $α$-spectral radius…
