Here we treat the problem of representability of an algebraic lat-
tice by the weak congruence lattice of an algebra, i.e. the lattice of all sym-
metric and transitive relations compatible with algebra. We prove that some
suborders of the representable lattices are representable, and give conditions
under which these suborders are also sublattices of the initial lattices. We
also prove that the direct product of a set of representable lattices, slightly
extended, is representable itself.
The rst and second multiplicative Zagreb indices of a graph G are
1(G) =
Σ
x2V (G)
d(x)2 and 2(G) =
Σ
(x;y)2E(G)
d(x) d(y), respectively, where
d(x) is the degree of the vertex x. We provide lower and upper bounds for 1
and 2 of a connected graph in terms of the number of vertices, number of
edges, and the ordinary, additive Zagreb indices M1 and M2
C.A. Barefoot, et. al. [6] introduced the concept of the integrity
of a graph. It is an useful measure of vulnerability and it is defined as follows.
I(G) = min fjSj + m(G
Each quasi-antiorder on anti-ordered semigroup S induces anti-
congruence q on S such that S=q is an ordered semigroup under anti-order
induced by . In this note we prove that the converse of this statement also
holds: Each anti-congruence q on a semigroup (S;=; ̸=; ·) such that S=q is an
anti-ordered semigroup induces a quasi-antiorder on S.
In this paper we define contractive bounded linear operators on
partially ordered Haussdorff topological vector space and study theirs basic
properties.
Let G = (V;E) be a simple graph. A partition of V (G) into in-
dependent,externally equitable sets is called externally equitable proper color
partition of G or externally equitable proper coloring of G. The minimum
cardinality of an externally equitable proper coloring of G is called exter-
nally equitable chromatic number of G and is denoted by ee(G). Since
= ffu1g; fu2g; ; fungg where V (G) = fu1; u2; ; ung is an externally
equitable proper coloring of G, externally equitable proper color partition ex-
ists in any graph G . In this paper, this new parameter is introduced and
studied.
We de ne a random iteration scheme and consider its convergence
to a common random xed point of two random operators de ned on a convex
subset of a separable Hilbert space.
The aim of this paper is to study the strong convergence of an
implicit iteration process to a common xed point for a nite family of asymp-
totically nonexpansive nonself mappings in a uniformly convex Banach spaces.
The aim of the present paper is to introduce the notion of weak
semi compatibility and obtain xed point theorems by employing the new
notion. The new notion is the proper generalization of semi compatibility
and is applicable to compatible and commuting type mappings. Our result
generalizes several xed point theorems.
In this paper we introduce and establish a cyclic contraction result
in probabilistic 2-metric spaces. A control function has been utilized in our
theorem. This result generalizes some existing results in 2-metric spaces. Our
result is illustrated with an example.
We analyse continuously differentiable functions f : Rr{0} → R,
that are the solutions of functional equation f(st) = f(s) + f(t): We prove
that f ≡ 0; and logarithmic functions f(t) = loga
|t|, (0 < a ̸= 1) are the only
solutions of the equation above.
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