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Let G be an irregular graph on n vertices with maximum degree ∆ and diameter D. We show that ∆ − λ 1 > 1 nD , where λ 1 is the largest eigenvalue of the adjacency matrix of G. We also study the effect of adding or removing few edges on the spectral radius of a regular graph. Our graph notation is standard (see West [22]). For a graph G, we denote by λ i (G)(More)
a r t i c l e i n f o a b s t r a c t Let G be a connected k-regular graph of order n. We find a best upper bound (in terms of k) on the third largest eigenvalue that is sufficient to guarantee that G has a perfect matching when n is even, and a matching of order n − 1 when n is odd. We also examine how other eigenvalues affect the size of matchings in G.
Let λ 1 be the greatest eigenvalue and λ n the least eigenvalue of the adjacency matrix of a connected graph G with n vertices, m edges and diameter D. We prove that if G is nonregular, then Δ − λ 1 > nΔ − 2m n(D(nΔ − 2m) + 1) 1 n(D + 1) , where Δ is the maximum degree of G. The inequality improves previous bounds of Stevanovi´c and of Zhang. It also(More)
The spectra of the skew-adjacency matrices of a graph are considered as a possible way to distinguish adjacency cospectral graphs. This leads to the following topics: graphs whose skew-adjacency matrices are all cospectral; relations between the matchings polynomial of a graph and the characteristic polynomials of its adjacency and skew-adjacency matrices;(More)
Let G be a connected graph. We study the extreme entries of the principal eigenvector x of G, the unique positive unit eigenvector corresponding to the greatest eigenvalue λ 1 of the adjacency matrix of G. If G has maximum degree ∆, we show that the greatest entry xmax of x is at most 1/ Õ 1 + λ 2 1 /∆. This improves a result of Papendieck and Recht. We(More)
We find lower bounds on the difference between the spectral radius λ 1 and the average degree 2e n of an irregular graph G of order n and size e. In particular, we show that, if n ≥ 4, then λ 1 − 2e n > 1 n(∆ + 2) where ∆ is the maximum of the vertex degrees in G. Brouwer and Haemers found eigenvalue conditions sufficient to imply the existence of perfect(More)
The minimum rank of a simple graph G is defined to be the smallest possible rank over all symmetric real matrices whose ijth entry (for i = j) is nonzero whenever {i, j} is an edge in G and is zero otherwise. This paper introduces a new graph parameter, Z(G), that is the minimum size of a zero forcing set of vertices and uses it to bound the minimum rank(More)