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Let G be a simple bipartite graph with bipartition (X, Y) such that |X|≥|Y|≥2. Put n=|X|, m=|Y| and p=m+n. Suppose, for each pair of nonadjacent vertices x∈X and y∈Y we have d(x)+d(y)≥n+1. We give an efficient algorithm to obtain a cycle of length 2m in G. The complexity of this algorithm is O(p2). In particular, if m=n, we have a hamiltonian cycle of G.

Let A A be a semisimple Banach algebra with ‖ ‖ \left \| \right \| , which is a dense subalgebra of a semisimple Banach algebra B B with | | \left | \right | such that ‖ ‖ \left \| \right \| majorizes | | \left | \right | on A A . The purpose of this paper is to investigate the dual property between the algebras A A and B B . Some well-known results follow from this paper.

We show that the algebra of double multipliers of a certain Banach algebra A A can be embedded in the second conjugate space A ∗ ∗ {A^{**}} of A A . This generalizes the previous work by the author for B ∗ {B^*} -algebras.

Let A be a dual B*-algebra and Ap the p-class in A. We show that the conjugate space of A1 is A**, the second conjugate space of A. We also obtain a three lines theorem for Ap (1 ≦p≦∞).

LetAbe a semisimple Banach algebra with ‖ · ‖, which is a dense subalgebra of a semisimple Banach algebraBwith | · | such that ‖ · ‖ majorises | · | onA. The purpose of this paper is to investigate the annihilator property between the algebrasAandB.

With the aid of a computer. we give a regular graph of girth 6 and valency 7, which has 90 vertices and show that this is the unique smallest graph with these properties.

Letf(v,5)be the number of vertices of a(v,5)-cage(v≥3). We give an upper bound forf(v,5)which is considerably better than the previously known upper bounds. In particular, whenv=7, it coincides with the well-known Hoffman- Singleton graph.

We study quasi-complementors and complementors on annihilator Banach algebras. We give conditions for a quasi-complementor to be a complementor.

Letf(11,6)be the number of vertices of an(11,6)-cage. By giving a regular graph of girth6and valency11, we show thatf(11,6)≤240. This is the best known upper bound forf(11,6).

Recently, O'Keefe and Wong have shown that a smallest graph of girth 10 and valency 3 (a (3, 10)-cage) must have 70 vertices. There are three non-isomorphic (3, 10)-cages which have been known for some while. In this paper, it is shown that these are the only three possible (3, 10)-cages. Some of the proofs in this paper are derived with the aid of a computer.

Let A be a Banach algebra and A** its second conjugate space. Arens has denned two natural extensions of the product on A to A**. Under either Arens product, A** becomes a Banach algebra. Let A be a semisimple Banach algebra which is a dense two-sided ideal of a B*-algebra B and R** the radical of (A**, o). We show that A** = Q ⊕ R**, where Q is a closed two-sided ideal of A**, o). This was inspired by Alexander's recent result for simple dual A*-algebras (see [1, p. 573, Theorem 5]). We also obtain that if A is commutative, then A is Arens regular.

Let A A be a semisimple Banach algebra and M ( A ) M(A) the algebra of double multipliers of A A . We show that M ( A ) M(A) is isomorphic to ( A ∗ ∗ , ∘ ) ({A^{ * * }}, \circ ) if and only if A A has the following properties: (1) A A is Arens regular, (2) A A has a weak approximate identity, and (3) π ( A ) \pi (A) is an ideal of ( A ∗ ∗ , ∘ ) ({A^{ * * }}, \circ ) .

Let A be a dual B ∗ {B^\ast } -algebra. We give a minimax formula for the positive elements in A. By using this formula and some of its consequent results, we introduce and study the symmetric norms and symmetrically-normed ideals in A.

The purpose of this paper is to generalize some results in a recent paper by Tomiuk and the author ([11]).

The study of complemented Banach*-algebras taken up in [1] was confined mainly toB*-algebras. In the present paper we extend this study to (right) complemented Banach*-algebras in whichx*x= 0 impliesx= 0. We show that ifAis such an algebra then every closed two-sided ideal ofAis a *-ideal. Using this fact we obtain a structure theorem forAwhich states that ifAis semi-simple thenAcan be expressed as a topological direct sum of minimal closed two sided ideals each of which is a complemented Banach*-algebra. It follows thatAis an A*-algebra and is a dense subalgebra of a dual B*-algebraU, which is determined uniquely up to *-isomorphism.

Let A be a modular complemented algebra. We give some necessary and sufficient conditions for some special classes of algebras A to be annihilator or dual algebras.

Let A A be a commutative B ∗ {B^ \ast } -algebra, Φ \Phi its carrier space and A ∗ {A^ \ast } the conjugate space of A A . Let A ′ A’ be the closed subspace of A ∗ {A^ \ast } spanned by Φ \Phi . We show that A A is a dual algebra if and only if A ′ = A ∗ A’ = {A^ \ast } and for each x ∈ A x \in A , the mapping T x : f → f ∗ x {T_x}:f \to f \ast x is a weakly completely continuous operator on A ∗ {A^ \ast } . This improves an early result by B. J. Tomiuk and the author. A similar result holds for general B ∗ {B^ \ast } -algebras.

In this paper, we study several problems in Banach algebras concerned with the Arens products.

The purpose of this paper is to generalize two results in a recent paper by B. J. Tomiuk and the author. Let A be a B ∗ {B^ \ast } -algebra and M ( A ) M(A) the algebra of double centralizers of A. We show that A is a dual algebra if and only if M ( A ) M(A) coincides with A ∗ ∗ {A^{ \ast \ast }} . We also obtain that if A is an annihilator A ∗ {A^\ast } -algebra, then π A ( A ) {\pi _A}(A) is a two-sided ideal of A ∗ ∗ {A^{ \ast \ast }} .

LetA1,A2be commutative semi-simple Banach algebras andM(A1),M(A2) their multiplier algebras. Birtel in [2] has proved that every isomorphism ofA1ontoA2induces an isomorphism ofM(A1) ontoM(A2). In this note, we extend this result to the noncommutative case. We also show that ifAis a dualA*-algebra which is a dense two-sided ideal of aB*-algebraB, thenM(A) is isomorphic toM(B). Thus the converse of the previous result cannot hold. All algebras under consideration are over the complex field.