Nonbacktracking spectrum of random graphs: Community detection and nonregular Ramanujan graphs

commuting and non -commuting graphs of finit groups

فرض کنیمg یک گروه غیر آبلی متناهی باشد . گراف جابجایی g که با نماد نمایش داده می شود ،گرافی است ساده با مجموعه رئوس که در آن دو راس با یک یال به هم وصل می شوند اگر و تنها اگر . مکمل گراف جابجایی g راگراف نا جابجایی g می نامیم.و با نماد نشان می دهیم. گرافهای جابجایی و ناجابجایی یک گروه متناهی ،اولین بار توسطاردوش1 مطرح گردید ،ولی در سالهای اخیر به طور مفصل در مورد بحث و بررسی قرار گرفتند . در ،م...

Ramanujan graphs

Ramanujan Graphs

In the last two decades, the theory of Ramanujan graphs has gained prominence primarily for two reasons. First, from a practical viewpoint, these graphs resolve an extremal problem in communication network theory (see for example [2]). Second, from a more aesthetic viewpoint, they fuse diverse branches of pure mathematics, namely, number theory, representation theory and algebraic geometry. The...

D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

The D-eigenvalues {µ1,…,µp} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, ED(G) of G is given by ED (G) =∑i=1p |µi|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated lin...

Extreme eigenvalues of nonregular graphs

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 an...