Largest 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.

The hypertree with the second largest spectral radius among uniform hypertrees with given size and stability number

The hypertree with the second largest spectral radius among uniform hypertrees with given size and stability number

Extensions on spectral extrema of C5/C6-free graphs with given size

Let F denote a set of graphs. A graph G is said to be F-free if it does not contain any element of F as a subgraph. The Turán number is the maximum possible number of edges in an F-free graph with n vertices. It is well known that classical Turán type extremal problem aims to study the Turán number of fixed graphs. In 2010, Nikiforov [17] proposed analogously a spectral Turán type problem which asks to determine the maximum spectral radius of an F-free graph with n vertices. It attracts much attention and many such problems remained elusive open even after serious attempts, and so they are considered as one of the most intriguing problems in spectral extremal graph theory. It is interesting to consider another spectral Turán type problem which asks to determine the maximum spectral radius of an F-free graph with m edges. Denote by G(m,F) the set of F-free graphs with m edges having no isolated vertices. Each of the graphs among G(m,F) having the largest spectral radius is called a maximal graph. Let θp,q,r be a theta graph formed by connecting two distinct vertices with three independent paths of length p,q and r, respectively (length refers to the number of edges). In this paper, we firstly determine the unique maximal graph among G(m,θ1,2,3) and G(m,θ1,2,4), respectively. Then we determine all the maximal graphs among G(m,C5) (resp. G(m,C6)) excluding the book graph. These results extend some earlier results.

The Hypertree with the Largest Spectral Radius Among all Hypertrees with Given Size and Stability Number

In this paper, we determine the unique uniform hypertree with the maximum spectral radius in the set (denoted by $${\cal T}(m,r,\alpha)$$ ) of all r-uniform hypertrees with given size m(> r) and stability number α. More precisely, for any $$T \in {\cal T}$$ with m > r, we have $$\rho (T) \le {\left({{1 \over {1 - {\alpha _0}}}} \right)^{1/r}},$$ where α0 is the maximum root of $${x^{r - 1}}\left({{1 \over {1 - x}} - {1 \over {{x^s}}} - l} \right) = g,$$ and integers g, l, s are determined by n − 1 − α = g(r − 1) + s, where n = m(r − 1) + 1 is the order of T and 0 ≤ s < r − 1, and l = m − gr − s − 1. Further, $$\rho (T) = {({1 \over {1 - {\alpha _0}}})^{1/r}}$$ if and only if T = R((r − 1)g; 0l; s).

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

Let G be a graph of size m and ρ(G) be the spectral radius of its adjacency matrix. A graph is said to be F-free if it does not contain a subgraph isomorphic to F. Let θ1,2,3 denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths 1, 2, 3, respectively. Recently, Li, Sun and Wei showed that for any θ1,2,3-free graph G of size m≥8, ρ(G)≤1+4m−32, with equality if and only if G≅Sm+32,2, where Sm+32,2=K2▿Km−12‾. However, this bound is not attainable when m is even. In this paper, we characterize the graphs with the maximum spectral radius over K2,r+1-free non-star graphs with m≥(4r+2)2+1, the graphs with the maximum spectral radius over θ1,2,3-free graphs with m≥22 when m is even, and the graphs with the second largest spectral radius over θ1,2,3-free graphs with m≥22 when m is odd.

Uniform supertrees with extremal spectral radii

A supertree is a connected and acyclic hypergraph. We investigate the supertrees with the extremal spectral radii among several kinds of r-uniform supertrees. First, by using the matching polynomials of supertrees, a new and useful grafting operation is proposed for comparing the spectral radii of supertrees, and its applications are shown to obtain the supertrees with the extremal spectral radii among some kinds of r-uniform supertrees. Second, the supertree with the third smallest spectral radius among the r-uniform supertrees is deduced. Third, among the r-uniform supertrees with a given maximum degree, the supertree with the smallest spectral radius is derived. At last, among the r-uniform starlike supertrees, the supertrees with the smallest and the largest spectral radii are characterized.

The Linear Unicyclic Hypergraph with the Second or Third Largest Spectral Radius

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.

The maximum [formula omitted]-spectral radius and the majorization theorem of [formula omitted]-uniform supertrees

Let α be a real number with 0≤α<1, G be a uniform hypergraph, and Aα(G)=αD(G)+(1−α)A(G), where D(G) and A(G) are the diagonal degree tensor and the adjacency tensor of G, respectively. The spectral radius of Aα(G) is called the α-spectral radius of G. In this paper, some properties for the α-spectral radius of connected k-uniform hypergraphs are investigated, the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with a given degree sequence is characterized, and the majorization theorem for k-uniform supertrees is obtained. By applying the majorization theorem, we determine the k-uniform supertrees obtaining the three largest α-spectral radius among all k-uniform supertrees with n vertices, give a new proof ordering the eight largest spectral radius among all k-uniform supertrees with n vertices, propose an open problem for further research, and characterize the unique k-uniform supertree with the largest α-spectral radius among all k-uniform supertrees with given different parameters, e.g., the maximum degree △, or the number of pendent vertices.

The second largest spectral radii of uniform hypertrees with given size of matching

ABSTRACT In this paper, we almost completely determine the hypertrees with the second largest spectral radius among all hypertrees with m edges and given size of matching by using the matching polynomials and ordering of hypertrees.

Spectral radii of two kinds of uniform hypergraphs

Let A(H) be the adjacency tensor (hypermatrix) of uniform hypergraph H. The maximum modulus of the eigenvalues of A(H) is called the spectral radius of H, denoted by ρ(H). In this paper, a conjecture concerning the spectral radii of linear bicyclic uniform hypergraphs is solved, with these results the hypergraph with the largest spectral radius is completely determined among the linear bicyclic uniform hypergraphs. For a t-uniform hypergraph G its generalized power r-uniform hypergraph Gr, s is defined in this paper. An exact relation between ρ(G) and ρ(Gr, s) is proved, more precisely ρ(Gr,s)=(ρ(G))tsr.

Ordering uniform supertrees by their spectral radii

A supertree is a connected and acyclic hypergraph. For a hypergraph H, the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H. By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on n vertices, which extends the known result.

The first few unicyclic and bicyclic hypergraphs with largest spectral radii

A connected k-uniform hypergraph with n vertices and m edges is called r-cyclic if n=m(k−1)−r+1. For r=1 or 2, the hypergraph is simply called unicyclic or bicyclic. In this paper we investigate hypergraphs that attain largest spectral radii among all simple connected k-uniform unicyclic and bicyclic hypergraphs. Specifically, by using some edge operations, the formula on power hypergraph eigenvalues, the weighted incidence matrix and a result on linear unicyclic hypergraphs, we determined the first five hypergraphs with largest spectral radius among all unicyclic hypergraphs and the first three over all bicyclic hypergraphs.

Trees with given maximum degree minimizing the spectral radius

The spectral radius of a graph is the largest eigenvalue of the adjacency matrix of the graph. Let $T^*(n,\Delta ,l)$ be the tree which minimizes the spectral radius of all trees of order $n$ with exactly $l$ vertices of maximum degree $\Delta $. In this paper, $T^*(n,\Delta ,l)$ is determined for $\Delta =3$, and for $l\le 3$ and $n$ large enough. It is proven that for sufficiently large $n$, $T^*(n,3,l)$ is a caterpillar with (almost) uniformly distributed legs, $T^*(n,\Delta ,2)$ is a dumbbell, and $T^*(n,\Delta ,3)$ is a tree consisting of three distinct stars of order $\Delta $ connected by three disjoint paths of (almost) equal length from their centers to a common vertex. The unique tree with the largest spectral radius among all such trees is also determined. These extend earlier results of Lov\' asz and Pelik\'an, Simi\' c and To\u si\' c, Wu, Yuan and Xiao, and Xu, Lin and Shu.

Ordering of some uniform supertrees with larger spectral radii

A supertree is a connected and acyclic hypergraph. We study some uniform supertrees with larger spectral radii. We first define a new type of edge-moving operation on hypergraphs, and study its applications on the comparison of the spectral radii of hypergraphs. By using this new operation on some supertrees together with the general edge-moving operation introduced by Li, Shao and Qi in 2015, and using a result by Zhou et al. in 2014 about the relation between the spectral radii of an ordinary graph and its power hypergraph, we are able to determine the first eight k-uniform supertrees on n vertices with the largest spectral radii.

Ordering n-vertex cacti with matching number q by their spectral radii

A connected graph G is a cactus if any two of its cycles have at most one common vertex. Denote by the set of n-vertex cacti with matching number q. Huang, Deng and Simić [23] identified the unique graph with the maximum spectral radius among 2q-vertex cacti with perfect matchings. In this paper, as a continuance of it, the largest and second largest spectral radii together with the corresponding graphs among are determined. Consequently, the first two largest spectral radii together with cacti having perfect matchings are also determined.

On graphs with cut vertices and cut edges

Let G (n, k, t) be a set of graphs with n vertices, k cut edges and t cut vertices. In this paper, we classify these graphs in G (n, k, t) according to cut vertices, and characterize the extremal graphs with the largest spectral radius in G (n, k, t).

Some results on the spectral radii of trees, unicyclic and bicyclic graphs

Let �(G), � for short, be the maximum degree of a graph G. In this paper, trees (resp., unicyclic graphs and bicyclic graphs), which attain the first and the second largest spectral radius with respect to the adjacency matrix in the class of trees (resp., unicyclic graphs and bicyclic graphs) with n vertices and the maximum degree �, where � � n+1 2 (resp., � � n +1 and � � n+3 ) are determined. Moreover, it is shown that the spectral radius of a unicyclic graph U (resp., a bicyclic graph B) on n vertices strictly increases with its maximum degree when �(U) � 1 1 + p 6n + 10 �2 (resp., �(B) � 1 2 + p 6n + 28 � 2 ).

On the spectral radius of weighted unicyclic graphs with a positive weight set

The spectrum of weighted graphs are often used to solve the problems in the design of networks and electronic circuits. In this article, we identify the (unique) weighted unicyclic graph with the largest spectral radius among the set of all n-vertex weighted unicyclic graphs with a fixed weight set W n = {w 1, w 2, … , w n }, where w 1 = w 2 = ··· = w n > 0 or, w 1 ⩾ w 2 ⩾ ··· ⩾ w n > 0 and .

On the spectral radii and the signless Laplacian spectral radii of c-cyclic graphs with fixed maximum degree

If G is a connected undirected simple graph on n vertices and n + c - 1 edges, then G is called a c-cyclic graph. Specially, G is called a tricyclic graph if c = 3 . Let Δ ( G ) be the maximum degree of G. In this paper, we determine the structural characterizations of the c-cyclic graphs, which have the maximum spectral radii (resp. signless Laplacian spectral radii) in the class of c-cyclic graphs on n vertices with fixed maximum degree Δ ⩾ n + c + 1 2 . Moreover, we prove that the spectral radius of a tricyclic graph G strictly increases with its maximum degree when Δ ( G ) ⩾ 1 + 6 + 2 n 3 2 , and identify the first six largest spectral radii and the corresponding graphs in the class of tricyclic graphs on n vertices.

ON THE SECOND LARGEST SPECTRAL RADIUS OF UNICYCLIC BIPARTITE GRAPHS

In this paper, we determine the two graphs in the class of unicyclic bipartite graphs with n vertices, having the first two largest spectral radius.