Extreme eigenvalues of nonregular graphs
نویسندگان
چکیده
Let λ1 be the greatest eigenvalue and λn 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− λ1 > 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 Stevanović and of Zhang. It also implies that a lower bound on λn obtained by Alon and Sudakov for (possibly regular) connected nonbipartite graphs also holds for connected nonregular graphs. © 2006 Elsevier Inc. All rights reserved.
منابع مشابه
Graphs with four distinct Laplacian eigenvalues
In this paper, we investigate connected nonregular graphs with four distinct Laplacian eigenvalues. We characterize all such graphs which are bipartite or have exactly one multiple Laplacian eigenvalue. Other examples of interest are also presented.
متن کاملMore on Non-regular Bipartite Integral Graphs with Maximum Degree 4 Not Having ±1 as Eigenvalues
A graph is integral if the spectrum (of its adjacency matrix) consists entirely of integers. The problem of determining all non-regular bipartite integral graphs with maximum degree four which do not have ±1 as eigenvalues was posed in K.T. Balińska, S.K. Simić, K.T. Zwierzyński: Which nonregular bipartite integral graphs with maximum degree four do not have ±1 as eigenvalues? Discrete Math., 2...
متن کاملConstructing Fifteen Infinite Classes of Nonregular Bipartite Integral Graphs
A graph is called integral if all its eigenvalues (of the adjacency matrix) are integers. In this paper, the graphs S1(t) = K1,t, S2(n, t), S3(m,n, t), S4(m,n, p, q), S5(m,n), S6(m,n, t), S8(m,n), S9(m,n, p, q), S10(n), S13(m,n), S17(m,n, p, q), S18(n, p, q, t), S19(m,n, p, t), S20(n, p, q) and S21(m, t) are defined. We construct the fifteen classes of larger graphs from the known 15 smaller in...
متن کاملSmall graphs with exactly two non-negative eigenvalues
Let $G$ be a graph with eigenvalues $lambda_1(G)geqcdotsgeqlambda_n(G)$. In this paper we find all simple graphs $G$ such that $G$ has at most twelve vertices and $G$ has exactly two non-negative eigenvalues. In other words we find all graphs $G$ on $n$ vertices such that $nleq12$ and $lambda_1(G)geq0$, $lambda_2(G)geq0$ and $lambda_3(G)0$, $lambda_2(G)>0$ and $lambda_3(G)
متن کاملOn the eigenvalues of normal edge-transitive Cayley graphs
A graph $Gamma$ is said to be vertex-transitive or edge- transitive if the automorphism group of $Gamma$ acts transitively on $V(Gamma)$ or $E(Gamma)$, respectively. Let $Gamma=Cay(G,S)$ be a Cayley graph on $G$ relative to $S$. Then, $Gamma$ is said to be normal edge-transitive, if $N_{Aut(Gamma)}(G)$ acts transitively on edges. In this paper, the eigenvalues of normal edge-tra...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- J. Comb. Theory, Ser. B
دوره 97 شماره
صفحات -
تاریخ انتشار 2007