In this paper, we give a necessary and sufficient condition for the integrality of Cayley graphs over the dihedral group D n = a, b | a n = b 2 = 1, bab = a −1 . Moreover, we also obtain some simple sufficient conditions for the integrality of Cayley graphs over D n in terms of the Boolean algebra of a , from which we find infinite classes of integral Cayley graphs over D n . In particular, we completely determine all integral Cayley graphs over the dihedral group D p for a prime p.

Let A n denote the alternating group of degree n with n ≥ 3. The alternating group graph AG n , extended alternating group graph EAG n and complete alternating group graph CAG n are the Cayley graphs Cay(A n , T 1 ), Cay(A n , T 2 ) and Cay(A n , T 3 ), respectively, whereIn this paper, we determine the second largest eigenvalues of AG n , EAG n and CAG n .

Let G(4,2) be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, G(4, 2, −1) (resp. G(4, 2, 0)) the set of graphs belonging to G(4, 2) with −1 (resp. 0) as an eigenvalue, and G(4, ≥ −1) the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than −1. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in G(4, 2, −1). As a by-product of this work, we characterize all the graphs belonging to G(4, ≥ −1) and G(4, 2, 0), respectively, and show that all these graphs are determined by their spectra.

Let Γ be a finite group acting transitively on [n] = {1, 2, . . . , n}, and let G = Cay(Γ, T ) be a Cayley graph of Γ. The graph G is called normal if T is closed under conjugation. In this paper, we obtain an upper bound for the second (largest) eigenvalue of the adjacency matrix of the graph G in terms of the second eigenvalues of certain subgraphs of G (see Theorem 2.6). Using this result, we develop a recursive method to determine the second eigenvalues of certain Cayley graphs of S n and we determine the second eigenvalues of a majority of the connected normal Cayley graphs (and some of their subgraphs) of S n with max τ ∈T |supp(τ )| ≤ 5, where supp(τ ) is the set of points in [n] non-fixed by τ .

Pemphigus is a rare autoimmune blistering disease, involving potentially life-threatening conditions often requiring immunosuppression. Currently, the COVID-19 pandemic caused by severe acute respiratory disease coronavirus 2 (SARS-CoV-2) infection has become a global public emergency. Vaccines are the most effective defense against COVID-19 infection. However, in clinic, there are cases of new onset or flare of pemphigus following COVID-19 vaccination, where vaccines have manifested significantly desirable risk-benefit profiles for patients. Although Rituximab, as first-line therapy, may impair humoral immunity, pemphigus may not predispose to develop COVID-19 infection compared to a healthy population. Conversely, delay or interruption of immunosuppressants probably results in unfavorable clinical outcomes for disease progression. Overall, clinicians should encourage their patients to undergo the vaccination after a comprehensive assessment. The definite association between COVID-19 vaccination and pemphigus remains to be further elucidated. Herein, we provide an overview of the published studies to date on COVID-19 and pemphigus as well as the exploration of their complicated interplay. In addition, we discuss the management strategies for pemphigus patients in this special period, in an effort to more effectively establish a standard treatment paradigm for this particular patient group.