Let [Formula: see text] be the set of all unbalanced unicyclic graphs of order [Formula: see text] and girth at least [Formula: see text]. In this paper, we determine the signed graph whose largest eigenvalue is maximal among all graphs in [Formula: see text].
Let G be a simple graph and A(G) be the adjacency matrix of G. The matrix S(G) = J − I − 2A(G) is called the Seidel matrix of G, where I is an identity matrix and J is a square matrix all of whose entries are equal to 1. Clearly, if G is a graph of order n with no isolated vertex, then the Seidel matrix of G is also the adjacency matrix of a signed complete graph Kn whose negative edges induce G. In this paper, we study the Seidel eigenvalues of the complete multipartite graph Kn 1 ,...,n k and investigate its Seidel characteristic polynomial. We show that if there are at least three parts of size ni, for some i = 1, . . . , k, then Kn 1 ,...,n k is determined, up to switching, by its Seidel spectrum.
Antihistamines play an important role in medicine when it comes to relieving seasonal or non-seasonal rhinitis, the common cold, and itching. They have also shown many various combinations of pharmacological properties such as anti-inflammatory and analgesic activities. Phenothiazines and ethylenediamines are 2 important classes of antihistamines with analgesic activities in addition to other pharmacological effects. In this study, some new derivatives of these compounds (V-IX) were synthesized and their antinociceptive behaviors were examined by pharmacological tests. The results indicated that new analogue with methyl groups produced a better analgesic activity than chlorine atoms but less than III (without any substitutions) in ethylenediamine class. Also in phenothiazine class, adding pyrimidine and pyridine substituted showed the better analgesic activity compared to other groups. Moreover, the analgesic activities proved that dimethylamine is the best group in amino alkyl side chain of these molecules relative to the substituted piperazines in new analogues.
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