The main aim of this paper is to extend the results of Nguyen
Van Luong and Nguyen Xuan Thuan [4] on xed point theory. Incidently, we
observed some inconsistencies in their proof and example. We extended their
results and exhibited a supporting example.
The concept of -reflexive semigroups is due to Chacron and
Thierrin in 1972. In the present paper we discuss -reflexive semigroups with
apartness. We prove some fundamental properties of these semigroups.
In this paper using Simpson's quadrature formula for convex func-
tion, the inequality involving ratio of means for four positive arguments proved
by Rooin and Hassni (2007) is re ned.
Let G = (V;E) be a graph, D ⊆ V and u be any vertex in D.
Then the out degree of u with respect to D denoted by odD(u), is de ned as
odD(u) = |N(u) ∩ (V − D)|. A subset D ⊆ V (G) is called a near equitable
dominating set of G if for every v ∈ V − D there exists a vertex u ∈ D
such that u is adjacent to v and |odD(u) − odV
The average lower domination number
av(G) is defined as
1
jV (G)jΣv2V (G)
v(G)
where
v(G) is the minimum cardinality of a maximal dominating set that
contains v. In this paper, the average lower domination number of complete
k-ary tree and Bn tree are calculated. Moreover we obtain the
av(G) for
thorn graph G. Finally we compute the
av(G1 + G2) of G1 and G2
The rst Zagreb index M1 of a graph G is equal to the sum of
squares of the vertex degrees of G. In a recent work [Goubko, MATCH Com-
mun. Math. Comput. Chem. 71 (2014), 33{46], it was shown that for a tree
with n1 pendent vertices, the inequality M1 > 9 n1
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