Let \(G=(V(G), E(G))\) be a connected graph and \(f: V(G) \rightarrow\{0,1,2\}\) be a hop Roman dominating function (HRDF) on \(G\). For each \(i \in\{0,1,2\}\), let \(V_{i}=\{v \in V(G): f(v)=i\}\). Then we write \(f=\) \(\left(V_{0}, V_{1}, V_{2}\right)\). A function \(f\) is a convex hop Roman dominating function (CvHRDF) on \(G\) if for every \(x \in V_{0}\), there exists \(y \in V_{2}\) such that \(d_{G}(x, y)=2\), and \(V_{1} \cup V_{2}\) is a convex set. The weight of CvHRDF \(f\) denoted by \(\omega_{G}^{C v h R}(f)\) is the sum \(\omega_{G}^{C v h R}(f)=\sum_{w \in V(G)} f(w)\). The convex hop Roman domination number of \(G\) denoted by \(\gamma_{C v h R}(G)\) is the minimum weight of a CvHRDF on \(G\). In this paper, we introduce and investigate a new parameter of hop Roman domination in graphs called the convex hop Roman dominating function. We also discuss important graph-theoretic results on the convex hop Roman domination number, including bounds, exact values, and characterizations.
This paper investigates the stabilization of two coupled onedimensional wave equations on adjacent intervals \((a, b)\) and \((b, c)\), connected by a dissipative interface at \(x=b\). A single dynamic boundary control with time delay acts at the free end \(x=c\) of the second interval, while the first equation satisfies a Dirichlet condition at \(x=a\) and is coupled to the second at \(x=b\) through continuity and dissipative conditions. By introducing an auxiliary variable to reformulate the delay, we express the system as an abstract Cauchy problem in a suitable Hilbert space. Under suitable conditions on the damping parameters, specifically \(e_{0}>e_{1}\) and \(\tau e_{1}
The concept of graph energy has attracted considerable attention due to its applications in spectral graph theory, combinatorics, chemistry, communication networks, and data science. In recent years, several variants of graph energy have been introduced by employing different graph matrices, including the adjacency matrix, Laplacian matrix, signless Laplacian matrix, and skew-adjacency matrix. Motivated by the skew Laplacian energy of digraphs, we introduce a new degree-based matrix called the skew maximum degree matrix. Using this matrix, we define the skew maximum degree Laplacian matrix of a digraph and study its associated energy. Let \(D(G)\) denote the diagonal matrix of vertex degrees and \(S(G)\) denote the skew maximum degree matrix. Then the skew maximum degree Laplacian matrix is defined by \[ L_{S}(G)=D(G)-S(G). \] The corresponding skew maximum degree Laplacian energy is introduced and investigated. Explicit expressions, spectral properties, lower and upper bounds, extremal results, and connections with classical graph invariants are established. The skew maximum degree Laplacian matrix appears to provide a promising bridge between classical skew spectral graph theory and modern degree-based approaches in mathematics.
We give an explicit negation orbit formulation of the standard Yang-Massey criterion for binary Euclidean LCD cyclic codes of odd length, counting both the zero code and the whole ambient space. With the defining set taken to be the full zero set of the generator polynomial, the LCD condition is \(T=-T\), so the enumeration is equivalent to choosing unions of orbits of the negation involution \(\imath(Q)=-Q\) on \(\mathcal{C}_{2, n}\), the set of 2-cyclotomic cosets modulo \(n\). The number of such codes is \[ 2^{\left|\mathcal{C}_{2, n} /\langle 1\rangle\right|}, \] and we record the corresponding divisor-sum expression in terms of \(\operatorname{ord}_{d}(2)\) and an explicit self-pairing indicator for each divisor \(d \mid n\). This is a binary cyclic specialization of known irreducible-factor enumerations and is used here only as setup. Our main result is a closed form enumeration at the Mersenne lengths \(n=2^{m}-1\), as a function of \(m\) alone. We prove \[ N_{L C D}(m)=2^{1+\frac{1}{2}(T(m)+S(m))}, \] where \(T(m)\) counts monic irreducible binary polynomials of degrees \(e>1\) dividing \(m\), and \(S(m)\) counts the self-reciprocal ones; both are explicit Möbius sums over the divisors of \(m\). In particular, the count requires no factorization of \(2^{m}-1\), and it extends the odd-prime- \(m\) formula of Li et al. [9] to all \(m \geq 2\). The exponent satisfies \(\log _{2} N_{L C D}(m)=2^{m-1} / m(1+o(1))\). For the lengths \(n=2^{m}+1\), where every binary cyclic code is LCD, the number of (necessarily LCD) binary cyclic codes is \(2^{1+F(m)}\) for an explicit Möbius sum \(F(m)\). The same exponent appears in the \(q=2\) case of the enumeration by Liu et al. [11] at the lengths \(\lambda\left(q^{m}+1\right)\), under a convention that omits the ambient space code; we state this overlap explicitly. For \(m \in\{3,4,5,6,7\}\), the Mersenne counts are 4, 16, 16, 256, 1024, and the closed form is verified against direct negation orbit enumeration for \(m \leq 16\).
We introduce and investigate the concepts of certified 2-dominating sets and certified 2-domination numbers of a graph. Here, we give necessary and sufficient conditions for a graph to attain a specific value of the parameter. It is shown that the difference \(\gamma_{2 \text { cer }}(G)-\gamma_{\text {cer }}(G)\) can be increased without bound, where \(\gamma_{2 \text { cer }}(G)\) and \(\gamma_{c e r}(G)\) denote the certified 2-domination and certified domination numbers of \(G\), respectively. Some results on the concept for the join of two non-trivial graphs are also obtained.
