The Markov graph of a self-map on a combinatorial tree is a directed graph that encodes the covering relations between edges of the tree under the map. This work explores the dynamical structure of self-maps on trees with weakly connected Markov graphs. The main result of the paper is a complete characterization of self-maps on finite sets that yield weakly connected Markov graphs for all trees. Additionally, we describe the dynamical structure of self-maps whose Markov graphs take specific forms, including complete digraphs, cycles, paths, in-stars, and out-stars.

In this paper, we construct two infinite families of graphs \(G(d,c)\) and \(G^+(d,c)\), where, in both cases, a vertex label is \(x_1x_2\ldots x_c\) with \(x_i\in\{1,2,\ldots, d\}\). We provide a tight lower bound on the metric dimension of \(G^+(d, c)\). Moreover, we give the definition and properties of the supertoken graphs, a generalization of the well-known token graphs. Finally, we provide an upper bound on the metric dimension of supertoken graphs.

Let \(A\) be a commutative ring with nonzero identity and \(n\geq 2\) be a positive integer. With the ring \(R=A\times\cdots\times A\) (\(n\) times), one can associate graphs \(TD(R)\) and \(ZD(R)\) respectively called the total dot product graph and the zero-divisor dot product graph of \(R\). In this paper, we study some topological properties of these two dot product graphs of \(R.\) In particular, it is shown that, the zero-divisor dot product graph \(ZD(R)\) is a projective graph if and only if \(R\) is isomorphic to \(\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}\times\frac{Z_2\left[x\right]}{\left\langle x^2+x+1\right\rangle}.\) Moreover, we prove that no total dot product graph can be projective. With these observations, we classify all commutative rings for which dot product graphs \(ZD(R)\) and \(TD(R)\) have crosscap two.

Some methods of decomposing \(v(=mn)\times b\) incidence matrix of regular group divisible (RGD) designs into square submatrices of order \(m\) are described. Such designs are known as tactical decomposable designs. As a by–product, resolvable solutions of some RGD designs are obtained. A relationship between tactical decomposable designs and \(\left(2,\ n\right)-\)threshold schemes is also given.

Let \(G=(V,E,F)\) be a planar graph with vertex set \(V\), edge set \(E\), and set of faces \(F.\) For nonnegative integers \(a,b,\) and \(c\), a type \((a,b,c)\) face-magic labeling of \(G\) is an assignment of \(a\) labels to each vertex, \(b\) labels to each edge, and \(c\) labels to each face from the set of integer labels \(\{1,2,\dots a|V|+b|E|+c|F|\}\) such that each label is used exactly once, and for each \(s\)-sided face \(f \in F,\) the sum of the label of \(f\) with the labels of the vertices and edges incident with \(f\) is equal to some fixed constant \(\mu_s\) for every \(s.\) We find necessary and sufficient conditions for every quadruple \((a,b,c,n)\) such that the \(n\)-prism graph \(Y_n \cong K_2 \square C_n\) admits a face-magic labeling of type \((a,b,c)\).

A special type of algebraic intersection graph called the \(n\)-inordinate invariant intersection graph has been constructed based on the symmetric group, and its structural properties are studied in the literature. In this article, we discuss the different types of dominator coloring schemes of the \(n\)-inordinate invariant intersection graphs and their complements, \(n\)-inordinate invariant non-intersection graphs, by obtaining the required coloring pattern and determining the graph invariant associated with the coloring.

Let \(G\) be a connected graph and let \(d_G\) be the geodesic distance on \(V(G)\). The metric spaces \((V(G), d_{G})\) were characterized up to isometry for all finite connected \(G\) by David C. Kay and Gary Chartrand in 1965. The main result of this paper expands this characterization on infinite connected graphs. We also prove that every metric space with integer distances between its points admits an isometric embedding in \((V(G), d_G)\) for suitable \(G\).

MacMahon extensively studied integer compositions, including the notion of conjugation. More recently, Agarwal introduced \(n\)-color compositions and their cyclic versions were considered by Gibson, Gray, and Wang. In this paper, we develop and study a conjugation rule for cyclic \(n\)-color compositions. Also, for fixed \(\ell\), we identify and enumerate the subset of self-conjugate compositions of \(\ell\), as well as establish a bijection between these and the set of cyclic regular compositions of \(\ell\) with only odd parts.

The covering cover pebbling number, \(\sigma(G)\), of a graph \(G\), is the smallest number such that some distribution \(D \in \mathscr{K}\) is reachable from every distribution starting with \(\sigma(G)\) (or more) pebbles on \(G\), where \(\mathscr{K}\) is a set of covering distributions. In this paper, we determine the covering cover pebbling number for two families of graphs those do not contain any cycles.

Jeff Remmel introduced the concept of a \(\mathit{k}\)-11-representable graph in 2017. This concept was first explored by Cheon et al. in 2019, who considered it as a natural extension of word-representable graphs, which are exactly 0-11-representable graphs. A graph \(G\) is \(k\)-11-representable if it can be represented by a word \(w\) such that for any edge (resp., non-edge) \(xy\) in \(G\) the subsequence of \(w\) formed by \(x\) and \(y\) contains at most \(k\) (resp., at least \(k+1\)) pairs of consecutive equal letters. A remarkable result of Cheon et al. is that  any graph is 2-11-representable, while it is still unknown whether every graph is 1-11-representable. Cheon et al. showed that the class of 1-11-representable graphs is strictly larger than that of word-representable graphs, and they introduced a useful toolbox to study 1-11-representable graphs, which was extended by additional powerful tools suggested by Futorny et al. in 2024. In this paper, we prove that all graphs on at most 8 vertices are 1-11-representable hence extending the known fact that all graphs on at most 7 vertices are 1-11-representable. Also, we discuss applications of our main result in the study of multi-1-11-representation of graphs we introduce in this paper analogously to the notion of multi-word-representation of graphs suggested by Kenkireth and Malhotra in 2023.