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Recent studies show that self-attentions behave like low-pass filters (as opposed to convolutions) and enhancing their high-pass filtering capability improves model performance. Contrary to this idea, we investigate existing convolution-based models with spectral analysis and observe that improving the low-pass filtering in convolution operations also leads to performance improvement. To account for this observation, we hypothesize that utilizing optimal token mixers that capture balanced representations of both high- and low-frequency components can enhance the performance of models. We verify this by decomposing visual features into the frequency domain and combining them in a balanced manner. To handle this, we replace the balancing problem with a mask filtering problem in the frequency domain. Then, we introduce a novel token-mixer named SPAM and leverage it to derive a MetaFormer model termed as SPANet. Experimental results show that the proposed method provides a way to achieve this balance, and the balanced representations of both high- and low-frequency components can improve the performance of models on multiple computer vision tasks. Our code is available at $\href{https://doranlyong.github.io/projects/spanet/}{\text{https://doranlyong.github.io/projects/spanet/}}$., Comment: Accepted paper at ICCV 2023

The middle graph $M(G)$ of a graph $G$ is the graph obtained by subdividing each edge of $G$ exactly once and joining all these newly introduced vertices of adjacent edges of $G$. A perfect Roman dominating function on a graph $G$ is a function $f : V(G) \rightarrow \{0, 1, 2\}$ satisfying the condition that every vertex $v$ with $f(v)=0$ is adjacent to exactly one vertex $u$ for which $f(u)=2$. The weight of a perfect Roman dominating function $f$ is the sum of weights of vertices. The perfect Roman domination number is the minimum weight of a perfect Roman dominating function on $G$. In this paper, we give a characterization of middle graphs with equal Roman domination and perfect Roman domination numbers.

Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is a \textit{restrained dominating set} if every vertex in $V \setminus D$ is adjacent to a vertex in $D$ and to a vertex in $V \setminus D$. The \textit{restrained domination number}, denoted by $\gamma_r(G)$, is the smallest cardinality of a restrained dominating set of $G$. A function $f : V \rightarrow \{0, 1, 2\}$ is a \textit{restrained Italian dominating function} on $G$ if (i) for each vertex $v \in V$ for which $f(v)=0$, it holds that $\sum_{u \in N_G(v)} f(u) \geq 2$, (ii) the subgraph induced by $\{v \in V \mid f(v)=0 \}$ has no isolated vertices. The \textit{restrained Italian domination number}, denoted by $\gamma_{rI}(G)$, is the minimum weight taken over all restrained Italian dominating functions of $G$. It is known that $\gamma_r(G) \leq \gamma_{rI}(G) \leq 2\gamma_r(G)$ for any graph $G$. In this paper, we characterize the trees $T$ for which $\gamma_r(T) = \gamma_{rI}(T)$, and we also characterize the trees $T$ for which $\gamma_{rI}(T) = 2\gamma_r(T)$.

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

An \textit{Italian dominating function} on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. The \textit{weight} of an Italian dominating function $f$ is the value $\omega(f) = f(V(D)) = \sum_{u \in V(D)} f(u)$. The \textit{Italian domination number} of a digraph $D$, denoted by $\gamma_I(D)$, is the minimum taken over the weights of all Italian dominating functions on $D$. The \textit{Italian bondage number} of a digraph $D$, denoted by $b_I(D)$, is the minimum number of arcs of $A(D)$ whose removal in $D$ results in a digraph $D'$ with $\gamma_I(D') > \gamma_I(D)$. The \textit{Italian reinforcement number} of a digraph $D$, denoted by $r_I(D)$, is the minimum number of extra arcs whose addition to $D$ results in a digraph $D'$ with $\gamma_I(D') < \gamma_I(D)$. In this paper, we initiate the study of Italian bondage and reinforcement numbers in digraphs and present some bounds for $b_I(D)$ and $r_I(D)$. We also determine the Italian bondage and reinforcement numbers of some classes of digraphs.

An Italian dominating function on a digraph $D$ with vertex set $V(D)$ is defined as a function $f : V(D) \rightarrow \{0, 1, 2\}$ such that every vertex $v \in V(D)$ with $f(v) = 0$ has at least two in-neighbors assigned $1$ under $f$ or one in-neighbor $w$ with $f(w) = 2$. In this paper, we determine the exact values of the Italian domination numbers of some products of directed cycles.

The main goal of the paper is to establish a sufficient condition for a two-valenced association scheme to be schurian and separable. To this end, an analog of the Desargues theorem is introduced for a noncommutative geometry defined by the scheme in question. It turns out that if the geometry has enough many Desarguesian configurations, then under a technical condition the scheme is schurian and separable. This result enables us to give short proofs for known statements on the schurity and separability of quasi-thin and pseudocyclic schemes. Moreover, by the same technique we prove a new result: given a prime $p$, any $\{1,p\}$-scheme with thin residue isomorphic to an elementary abelian $p$-group of rank greater than two, is schurian and separable., Comment: 14 pages

A permutation group is said to be quasiregular if every its transitive constituent is regular, and a quasiregular coherent configuration can be thought as a combinatorial analog of such a group: the transitive constituents are replaced by the homogeneous components. In this paper, we are interested in the question when the configuration is schurian, i.e., formed by the orbitals of a permutation group, or/and separable, i.e., uniquely determined by the intersection numbers. In these terms, an old result of Hanna Neumann is, in a sense, dual to the statement that the quasiregular coherent configurations with cyclic homogeneous components are schurian. In the present paper, we (a) establish the duality in a precise form and (b) generalize the latter result by proving that a quasiregular coherent configuration is schurian and separable if the groups associated with homogeneous components have distributive lattices of normal subgroups.

In this paper we aim to characterize association schemes all of whose symmetric fusion schemes have only integral eigenvalues, and classify those obtained from a regular action of a finite group by taking its orbitals.

In this article, we consider the existence and schurity problem on commutative $p$-schemes of order $p^4$. Using the thin radical and thin residue, we give sufficient conditions for such $p$-schemes to be schurian. We also give questions related to our results.

Recently, it was proved that every commutative $p$-Schur ring over a group of order $p^3$ is Schurian. In this article, we consider the Schurity problem of non-commutative $p$-Schur rings over groups of order $p^3$. In particular, it is given a family of non-Schurian $p$-Schur rings over groups of order $p^3$.

In this article, we investigate the existence and schurity problem of association schemes whose thin residues are isomorphic to an elementary abelian $p$-group of rank $2$.

Recently G. Bhattacharyya, S.Y. Song and R. Tanaka began to study Terwilliger algebras of wreath products of one-class association schemes. K. Kim determined the structure of Terwilliger algebras of wreath products by one-class association schemes or quasi-thin schemes. In this paper, we study Terwilliger algebras of wreath products by $3$-equivalenced schemes., Comment: arXiv admin note: substantial text overlap with arXiv:1109.1334

Every Hadamard matrix $H$ of order $n > 1$ induces a graph with $4n$ vertices, called the Hadamard graph $\Gamma(H)$ of $H$. Since $\Gamma(H)$ is a distance-regular graph with diameter $4$, it induces a $4$-class association scheme $(\Omega, S)$ of order $4n$. In this article we deal with fission schemes of $(\Omega, S)$ under certain conditions, and for such a fission scheme we estimate the number of isomorphism classes with the same intersection numbers as the fission scheme., Comment: 12 pages

Let $(X,S)$ be a $p$-scheme of order $p^3$ and $T$ the thin residue of $S$. Now we assume that $T$ has valency $p^2$. It is easy to see that one of the following holds: (i) $|T|=p^2$ and $T\simeq C_{p^2}$; (ii) $|T|=p^2$ and $T\simeq C_p\times C_p$; (iii) $|T|

The structure of Terwilliger algebras of wreath products by thin schemes or one-class schemes was studied in [A. Hanaki, K. Kim, Y. Maekawa, Terwilliger algebras of direct and wreath products of association schemes, J. Algebra 343 (2011) 195--200]. In this paper, we will consider the structure of Terwilliger algebras of wreath products by quasi-thin schemes. This gives a generalization of their result.

In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that strongly regular graphs constructed from copolar spaces and from the more general spaces called $\Delta$-spaces are counterexamples to Brouwer's Conjecture. Using J.I. Hall's characterization of finite reduced copolar spaces, we find that the triangular graphs $T(m)$, the symplectic graphs $Sp(2r,q)$ over the field $\mathbb{F}_q$ (for any $q$ prime power), and the strongly regular graphs constructed from the hyperbolic quadrics $O^{+}(2r,2)$ and from the elliptic quadrics $O^{-}(2r,2)$ over the field $\mathbb{F}_2$, respectively, are counterexamples to Brouwer's Conjecture. For each of these graphs, we determine precisely the minimum number of vertices whose removal disconnects the graph into non-singleton components. While we are not aware of an analogue of Hall's characterization theorem for $\Delta$-spaces, we show that complements of the point graphs of certain finite generalized quadrangles are point graphs of $\Delta$-spaces and thus, yield other counterexamples to Brouwer's Conjecture. We prove that Brouwer's Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles $GQ(q,q)$ graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue -2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases. We leave as an open problem determining the best general lower bound for the minimum size of a disconnecting set of vertices of a strongly regular graph, whose removal disconnects the graph into non-singleton components., Comment: 25 pages, 1 table; accepted to JCTA; revised version contains a new section on copolar and Delta spaces

We describe explicitly the algebraic structure of the Terwilliger algebra of wreath products of cyclic schemes.