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### Volume 63, number 1 (2022)

June 2022
 Front matter Spectral distances in some sets of graphs. Irena M. Jovanović Some of the spectral distance related parameters (cospectrality, spectral eccentricity, and spectral diameter with respect to an arbitrary graph matrix) are determined in one particular set of graphs. According to these results, the spectral distances connected with the adjacency matrix and the corresponding distance related parameters are computed in some sets of trees. Examples are provided of graphs whose spectral distances related to the adjacency matrix, the Laplacian and the signless Laplacian matrix are mutually equal. The conjecture related to the spectral diameter of the set of connected regular graphs with respect to the adjacency matrix is disproved using graph energy. 1–20 Approximation via statistical $K_{a}^{2}$-convergence on two-dimensional weighted spaces. Sevda Yıldız We give a non-regular statistical summability method named statistical $K_{a}^{2}$-convergence and prove a Korovkin type approximation theorem for this new and interesting convergence method on two-dimensional weighted spaces. We also study the rate of statistical $K_{a}^{2}$-convergence by using the weighted modulus of continuity and afterwards we present a non-trivial application. 21–39 Some relations between the skew spectrum of an oriented graph and the spectrum of certain closely associated signed graphs. Zoran Stanić Let $R_{G'}$ be the vertex-edge incidence matrix of an oriented graph $G'$. Let $\Lambda(\dot{F})$ be the signed graph whose vertices are identified as the edges of a signed graph $\dot{F}$, with a pair of vertices being adjacent by a positive (resp. negative) edge if and only if the corresponding edges of $\dot{G}$ are adjacent and have the same (resp. different) sign. In this paper, we prove that $G'$ is bipartite if and only if there exists a signed graph $\dot{F}$ such that $R_{G'}^\intercal R_{G'}-2I$ is the adjacency matrix of $\Lambda(\dot{F})$. It occurs that $\dot{F}$ is fully determined by $G'$. As an application, in some particular cases we express the skew eigenvalues of $G'$ in terms of the eigenvalues of $\dot{F}$. We also establish some upper bounds for the skew spectral radius of $G'$ in both the bipartite and the non-bipartite case. 41–50 The convex and weak convex domination number of convex polytopes. Zoran Lj. Maksimović, Aleksandar Lj. Savić, and Milena S. Bogdanović This paper is devoted to solving the weakly convex dominating set problem and the convex dominating set problem for some classes of planar graphs—convex polytopes. We consider all classes of convex polytopes known from the literature and present exact values of weakly convex and convex domination number for all classes, namely $A_n$, $B_n$, $C_n$, $D_n$, $E_n$, $R_n$, $R''_n$, $Q_n$, $S_n$, $S''_n$, $T_n$, $T''_n$ and $U_n$. When $n$ is up to 26, the values are confirmed by using the exact method, while for greater values of $n$ theoretical proofs are given. 51–68 The $\mathfrak{A}$-principal real hypersurfaces in complex quadrics. Tee-How Loo A real hypersurface in the complex quadric $Q^m=SO_{m+2}/SO_mSO_2$ is said to be $\mathfrak{A}$-principal if its unit normal vector field is singular of type $\mathfrak{A}$-principal everywhere. In this paper, we show that a $\mathfrak{A}$-principal Hopf hypersurface in $Q^m$, $m\geq3$, is an open part of a tube around a totally geodesic $Q^{m+1}$ in $Q^m$. We also show that such real hypersurfaces are the only contact real hypersurfaces in $Q^m$. The classification for complete pseudo-Einstein real hypersurfaces in $Q^m$, $m\geq3$, is also obtained. 69–91 On the module intersection graph of ideals of rings. Thangaraj Asir, Arun Kumar, and Alveera Mehdi Let $R$ be a commutative ring and $M$ an $R$-module. The $M$-intersection graph of ideals of $R$ is an undirected simple graph, denoted by $G_{M}(R)$, whose vertices are non-zero proper ideals of $R$ and two distinct vertices are adjacent if and only if $IM\cap JM\neq 0$. In this article, we focus on how certain graph theoretic parameters of $G_M(R)$ depend on the properties of both $R$ and $M$. Specifically, we derive a necessary and sufficient condition for $R$ and $M$ such that the $M$-intersection graph $G_M(R)$ is either connected or complete. Also, we classify all $R$-modules according to the diameter value of $G_M(R)$. Further, we characterize rings $R$ for which $G_M(R)$ is perfect or Hamiltonian or pancyclic or planar. Moreover, we show that the graph $G_{M}(R)$ is weakly perfect and cograph. 93–107 On Baer modules. Chillumuntala Jayaram, Ünsal Tekir, and Suat Koç A commutative ring $R$ is said to be a Baer ring if for each $a\in R$, $\operatorname{ann}(a)$ is generated by an idempotent element $b\in R$. In this paper, we extend the notion of a Baer ring to modules in terms of weak idempotent elements defined in a previous work by Jayaram and Tekir. Let $R$ be a commutative ring with a nonzero identity and let $M$ be a unital $R$-module. $M$ is said to be a Baer module if for each $m\in M$ there exists a weak idempotent element $e\in R$ such that $\operatorname{ann}_{R}(m)M=eM$. Various examples and properties of Baer modules are given. Also, we characterize a certain class of modules/submodules such as von Neumann regular modules/prime submodules in terms of Baer modules. 109–128 A characterization of Stone and linear Heyting algebras. Alejandro Petrovich and Carlos Scirica An important problem in the variety of Heyting algebras $\mathcal{H}$ is to find new characterizations which allow us to determinate if a given $H\in \mathcal{H}$ is linear or Stone. In this work we present two Heyting algebras, $H^{ns}$ and $H^{snl}$, such that: (a) a Heyting algebra $H$ is a Stone–Heyting algebra if and only if $H^{ns}$ cannot be embedded in $H$, and (b) $H$ is a linear Heyting algebra if and only if neither $H^{ns}$ nor $H^{snl}$ can be embedded in $H$. 129–136 Positive periodic solutions of a discrete ratio-dependent predator-prey model with impulsive effects. Cosme Duque and José L. Herrera Diestra Studies of the dynamics of predator-prey systems are abundant in the literature. In an attempt to account for more realistic models, previous studies have opted for the use of discrete predator-prey systems with ratio-dependent functional response. In the present research we go a step further with the inclusion of impulsive effects in the dynamics. More concretely, by assuming that the coefficients involved in the system and the impulses are periodic, we obtain sufficient conditions for the existence of periodic solutions. We present some numerical examples to illustrate the effectiveness of our results. 137–151 The total co-independent domination number of some graph operations. Abel Cabrera Martínez, Suitberto Cabrera García, Iztok Peterin, and Ismael G. Yero A set $D$ of vertices of a graph $G$ is a total dominating set if every vertex of $G$ is adjacent to at least one vertex of $D$. The total dominating set $D$ is called a total co-independent dominating set if the subgraph induced by $V(G) - D$ is edgeless. The minimum cardinality among all total co-independent dominating sets of $G$ is the total co-independent domination number of $G$. In this article we study the total co-independent domination number of the join, strong, lexicographic, direct and rooted products of graphs. 153–168 On the image set and reversibility of shift morphisms over discrete alphabets. Jorge Campos, Neptalí Romero, and Ramón Vivas We provide sufficient conditions in order to show that the image set of a continuous and shift-commuting map defined on a shift space over an arbitrary discrete alphabet is also a shift space. Additionally, if such a map is injective, then its inverse is also continuous and shift-commuting. 169–184 $C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces. Abdelkhalek El Amrani, Aziz Blali, and Jawad Ettayb We introduce and study $C$-groups and mixed $C$-groups of bounded linear operators on non-archimedean Banach spaces. Our main result extends some existing theorems on this topic. In contrast with the classical setting, the parameter of a given $C$-group (or mixed $C$-group) belongs to a clopen ball $\Omega_{r}$ of the ground field $\mathbb{K}$. As an illustration, we discuss the solvability of some homogeneous $p$-adic differential equations for $C$-groups and inhomogeneous $p$-adic differential equations for mixed $C$-groups when $\alpha=-1$. Examples are given to support our work. 185–201 Stability conditions and maximal green sequences in abelian categories. Thomas Brüstle, David Smith, and Hipolito Treffinger We study the stability functions on abelian categories introduced by Rudakov and their relation with torsion classes and maximal green sequences. Moreover, we introduce the concept of red paths, a stability condition in the sense of Rudakov that captures information of the wall and chamber structure of the category. 203–221 Trees with a unique maximum independent set and their linear properties. Daniel A. Jaume, Gonzalo Molina, and Rodrigo Sota Trees with a unique maximum independent set encode the maxi-mum matching structure in every tree. In this work we study some of their linear properties and give two graph operations, stellare and S-coalescence, which allow building all trees with a unique maximum independent set. The null space structure of any tree can be understood in terms of these graph operations. 223–238 New uncertainty principles for the $(k,a)$-generalized wavelet transform. Hatem Mejjaoli We present the basic $(k,a)$-generalized wavelet theory and prove several Heisenberg-type inequalities for this transform. After reviewing Pitt's and Beckner's inequalities for the $(k,a)$-generalized Fourier transform, we connect both inequalities to show a generalization of uncertainty principles for the $(k,a)$-generalized wavelet transform. We also present two concentration uncertainty principles, namely the Benedicks–Amrein–Berthier's uncertainty principle and local uncertainty principles. Finally, we connect these inequalities to show a generalization of the uncertainty principle of Heisenberg type and we prove the Faris–Price uncertainty principle for the $(k,a)$-generalized wavelet transform. 239–279 A Clemens–Schmid type exact sequence over a local basis. Genaro Hernandez-Mada Let $k$ be a finite field of characteristic $p$ and let $X \rightarrow \operatorname{Spec} k[[t]]$ be a semistable family of varieties over $k$. We prove that there exists a Clemens–Schmid type exact sequence for this family. We do this by constructing a larger family defined over a smooth curve and using a Clemens–Schmid exact sequence in characteristic $p$ for this new family. 281–292