Let G be a random subgraph of the n-cube where each edge appears randomly and independently with probability p. We prove that the largest eigenvalue of the adjacency matrix of G is almost surely λ1(G) = (1 + o(1))max ( 1/2(G), np ) , where (G) is the maximum degree ofG and theo(1) term tends to zero as max( 1/2(G), np) tends to infinity.

let $n$ be any positive integer and let $f_n$ be the friendship (or dutch windmill) graph with $2n+1$ vertices and $3n$ edges. here we study graphs with the same adjacency spectrum as the $f_n$. two graphs are called cospectral if the eigenvalues multiset of their adjacency matrices are the same. let $g$ be a graph cospectral with $f_n$. here we prove that if $g$ has no cycle of length $4$ or $...

We report the synthesis, x-ray diffraction, magnetic susceptibility and specific heat measurements on polycrystalline samples of undoped LiNi(2)P(3)O(10) and samples with non-magnetic impurity (Zn(2+), S = 0) and magnetic impurity (Cu(2+), S = 1/2) at the Ni site. The magnetic susceptibility data show a broad maximum at around 10 K and a small anomaly at about 7 K in the undoped sample. There i...

In this paper, it is shown that among connected graphs with maximum clique size ω, the minimum value of the spectral radius of adjacency matrix is attained for a kite graph PKn−ω,ω , which consists of a complete graph Kω to a vertex of which a path Pn−ω is attached. For any fixed ω, a small interval to which the spectral radii of kites PKm,ω , m ≥ 1, belong is exhibited.

Let G be a graph with n vertices and μ (G) be the largest eigenvalue of the adjacency matrix of G. We study how large μ (G) can be when G does not contain cycles and paths of specified order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of ...

The spectral radius of connected non-regular graphs is considered. Let λ1 be the largest eigenvalue of the adjacency matrix of a graph G on n vertices with maximum degree ∆. By studying the λ1-extremal graphs, it is proved that if G is non-regular and connected, then ∆− λ1 > ∆+ 1 n(3n+∆− 8) . This improves the recent results by B.L. Liu et al. AMS subject classifications. 05C50, 15A48.

Let G be a graph with n vertices and (G) be the largest eigenvalue of the adjacency matrix of G: We study how large (G) can be when G does not contain cycles and paths of speci…ed order. In particular, we determine the maximum spectral radius of graphs without paths of given length, and give tight bounds on the spectral radius of graphs without given even cycles. We also raise a number of natur...