Bulletin of the Malaysian Mathematical Sciences Society最新文献

Let (G_S) be the graph obtained by attaching a self-loop at every vertex in (S subseteq V(G)) of a simple graph G of order n. In this paper, we explore several new results related to the line graph (L(G_S)) of (G_S.) Particularly, we show that every eigenvalue of (L(G_S)) must be at least (-2,) and relate the characteristic polynomial of the line graph L(G) of G with the characteristic polynomial of the line graph (L({widehat{G}})) of a self-loop graph ({widehat{G}}), which is obtained by attaching a self-loop at each vertex of G. Then, we provide some new bounds for the eigenvalues and energy of (G_S.) As one of the consequences, we obtain that the energy of a connected regular complete multipartite graph is not greater than the energy of the corresponding self-loop graph. Lastly, we establish a lower bound of the spectral radius in terms of the first Zagreb index (M_1(G)) and the minimum degree (delta (G),) as well as proving two Nordhaus–Gaddum-type bounds for the spectral radius and the energy of (G_S,) respectively.

For any non-negative integer k and any graph G, a subset (Ssubseteq V(G)) is said to be a (K_{1,k+1})-isolating set of G if (G-N[S]) does not contain (K_{1,k+1}) as a subgraph. The (K_{1,k+1})-isolation number of G, denoted by (iota _k(G)), is the minimum cardinality of a (K_{1,k+1})-isolating set of G. Recently, Zhang and Wu (2021) proved that if G is a connected n-vertex graph and (Gnotin {P_3,C_3,C_6}), then (iota _1(G)le frac{2}{7}n). In this paper, we characterize all extremal graphs attaining this bound, which resolves a problem proposed by Zhang and Wu (Discrete Appl Math 304:365–374, 2021).

Let (mathbb {A}) be the (2times 2) diagonal operator matrix determined by a positive Hilbert space operator A. We give several upper bounds for the (mathbb {A})-Berezin number of (2times 2) block matrices on a reproducing kernel Hilbert space and prove inequalities for the A-Berezin number of Hilbert space operators. Our results in this paper generalize and refine earlier the A-Berezin number inequalities.

In this paper, we consider the following nonlinear Schrödinger equation with an (L^2)-constraint:

where (Nge 3), (a,mu >0), (2<q<2+frac{4}{N}<p<2^*), (2q+2N-pN<0) and (lambda in mathbb {R}) arises as a Lagrange multiplier. We deal with the concave and convex cases of energy functional constraints on the (L^2) sphere, and prove the existence of infinitely solutions with positive energy levels.

We study nuclear embeddings for spaces of Morrey type, both in its sequence space version and as smoothness spaces of functions defined on a bounded domain (Omega subset {{mathbb {R}}}^{{d}}). This covers, in particular, the meanwhile well-known and completely answered situation for spaces of Besov and Triebel-Lizorkin type defined on bounded domains which has been considered for a long time. The complete result was obtained only recently. Compact embeddings for function spaces of Morrey type have already been studied in detail, also concerning their entropy and approximation numbers. We now prove the first and complete nuclearity result in this context. The concept of nuclearity has already been introduced by Grothendieck in 1955. Again we rely on suitable wavelet decomposition techniques and the famous Tong result (1969) which characterises nuclear diagonal operators acting between sequence spaces of (ell _r) type, (1le rle infty ).

Given a simple, finite graph with vertex set V(G), we define a zero forcing set of G as follows. Choose (Ssubseteq V(G)) and color all vertices of S blue and all vertices in (V(G) - S) white. The color change rule is if w is the only white neighbor of blue vertex v, then we change the color of w from white to blue. If after applying the color change rule as many times as possible eventually every vertex of G is blue, we call S a zero forcing set of G. Z(G) denotes the minimum cardinality of a zero forcing set. We show that if G is 2-edge-connected, claw-free, and cubic, then . We also study a similar graph invariant known as the loop zero forcing number of a graph G which happens to be the dual invariant to the Grundy domination number of G. Specifically, we study the loop zero forcing number in two particular types of planar graphs.

The t-path ideal (I_t(G)) of a graph G is the square-free monomial ideal generated by the monomials which correspond to the paths of length t in G. In this paper, we prove that the Stanley–Reisner complex of the 2-path ideal (I_2(G)) of an (undirected) tree G is vertex decomposable. As a consequence, we show that the Alexander dual (I_2(G)^{vee }) of (I_2(G)) has linear quotients. For each (t ge 3), we provide a counterexample of a tree for which the Stanley–Reisner complex of (I_t(G)) is not vertex decomposable.

In this article, we investigate a class of nonsmooth multiobjective mathematical optimization problems with switching constraints (abbreviated as, (NMMPSC)) in the framework of Hadamard manifolds. Corresponding to (NMMPSC), the generalized Guignard constraint qualification (abbreviated as, (GGCQ)) is introduced in the Hadamard manifold setting. Karush–Kuhn–Tucker (abbreviated as, KKT) type necessary conditions of Pareto efficiency are derived for (NMMPSC). Subsequently, we introduce several other constraint qualifications for (NMMPSC), which turn out to be sufficient conditions for (GGCQ). We have furnished non-trivial illustrative examples to justify the significance of our results. To the best of our knowledge, constraint qualifications for (NMMPSC) have not yet been studied in the Hadamard manifold framework.

In this paper, we present a probabilistic extension of the Fubini polynomials and numbers associated with a random variable satisfying some appropriate moment conditions. We obtain the exponential generating function and an integral representation for it. The higher order Fubini polynomials and recurrence relations are also derived. A probabilistic generalization of a series transformation formula and some interesting examples are discussed. A connection between the probabilistic Fubini polynomials and Bernoulli, Poisson, and geometric random variables are also established. Finally, a determinant expression formula is presented.

We study the long-time dynamics of a wave equation with nonlocal weak damping, nonlocal weak anti-damping and sup-cubic nonlinearity. Based on the Strichartz estimates in a bounded domain, we obtain the global well-posedness of the Shatah–Struwe solutions. To overcome the difficulties brought by the nonlinear weak damping term, we present a new-type Gronwall’s lemma to obtain the dissipative for the Shatah–Struwe solutions semigroup of this equation. Finally, we establish the existence of a time-dependent exponential attractor with the help of a more general criteria constructed by the quasi-stable technique.