We consider the set Gn,k of graphs of order n with the chromatic number k ≥ 2. In this note, we prove that in Gn,k the Turán graph Tn,k has the maximal spectral radius; and Pn if k = 2, Cn if k = 3 and n is odd, C1 n−1 if k = 3 and n is even, K (l) k if k ≥ 4 has the minimal spectral radius. Thus we answer a problem raised by Cao [D.S. Cao, Index function of graphs, J. East China Norm. Univ. Sc...

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The greatest eigenvalue λ1 of D(G) is called the distance spectral radius of the graph G and is denoted by λ1(G). A simple connected graph G is called a 2-partite distance regular graph if there exists a partition V1 ∪V2 of the vertex set of G such that for i = 1, 2 and an...

The spectral radii of refinement and subdivision operators considered on the space L2 can be estimated by using norms of their symbols. In several cases, including those arising in wavelet analysis, the exact value of the spectral radius is found. For example, if T is the unit circle and if the symbol a of a refinement operator satisfies the conditions |a(z)|2 + |a(−z)|2 = 4, z ∈ T, and a(1) = ...

We give suucient conditions to approximate the \nonessential" spectrum of a bounded operator L acting on a Banach space B by part of the spectra of a sequence of compact (or nite rank) operators L j = (Id ? j)L(Id ? j), where Id ? j is a suitable family of uniformly bounded operators which approach the identity. (By nonessential spectrum we mean here all the spectrum outside of the disc of radi...

The independence number of a graph is defined as the maximum size of a set of pairwise non-adjacent vertices and the spectral radius is defined as the maximum eigenvalue of the adjacency matrix of the graph. Xu et al. in [The minimum spectral radius of graphs with a given independence number, Linear Algebra and its Applications 431 (2009) 937–945] determined the connected graphs of order n with...

it is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. in this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.

let $n$ be any positive integer, the friendship graph $f_n$ consists of $n$ edge-disjoint triangles that all of them meeting in one vertex. a graph $g$ is called cospectral with a graph $h$ if their adjacency matrices have the same eigenvalues. recently in href{http://arxiv.org/pdf/1310.6529v1.pdf}{http://arxiv.org/pdf/1310.6529v1.pdf} it is proved that if $g$ is any graph cospectral with $f_n$...

let $g$ be a simple graph with vertex set $v(g) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2,cdots, n$. inspired by the randi'c matrix and the general randi'cindex of a graph, we introduce the concept of general randi'cmatrix $textbf{r}_alpha$ of $g$, which is defined by$(textbf{r}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ areadjacent, and zero otherwise. s...