The purpose of this note is the study of the hypergroups associated with binary relations. New types of matrices, called \(i\)-very good and regular reversible matrices, are introduced in order to give some properties of the Rosenberg hypergroups related to them. A program written in MATLAB computes the number of these hypergroups up to isomorphism.

Let \(A_n\) be the alternating group of degree \(n\) with \(n > 4\). Set \(T = \{(1 2 3), (1 3 2), (1 2)(3 i) \mid 4 \leq i \leq n\}\). The alternating group network, denoted by \(AN_n\), is defined as the Cayley graph on \(A_n\) with respect to \(T\). Some properties of \(AN_n\) have been investigated in [App. Math.—JCU, Ser. A 14 (1998) 235-239; IEEE Trans. Comput. 55 (2006) 1645-1648; Inform. Process. Lett. 110 (2010) 403-409; J. Supercomput. 54 (2010) 206-228]. In this paper, it is shown that the full automorphism group of \(AN_n\) is the semi-direct product \(R(A_n) \rtimes \text{Aut}(A_n, T)\), where \(R(A_n)\) is the right regular representation of \(A_n\) and \(\text{Aut}(A_n, T) = \{\alpha \in \text{Aut}(A_n) \mid T^\alpha = T\} \cong S_{n-3} \times S_2\).

The harmonic index of a graph \(G\) is defined as the sum of weights \(\frac{2}{\deg(v) + \deg(u)}\) of all edges \(uv\) in \(E(G)\), where \(\deg(v)\) denotes the degree of a vertex \(v\) in \(V(G)\). In this note, we generalize results of [L. Zhong, The harmonic index on graphs, Appl. Math. Lett. 25 (2012), 561-566] and establish some upper and lower bounds on the harmonic index of \(G\).

Let \(\lambda_1, \lambda_2, \ldots, \lambda_n\) be the eigenvalues of the distance matrix of a connected graph \(G\). The distance Estrada index of \(G\) is defined as \(DEE(G) = \sum_{i=1}^{n} e^{\lambda_i}\). In this note, we present new lower and upper bounds for \(DEE(G)\). In addition, a Nordhaus-Gaddum type inequality for \(DEE(G)\) is given.

The splitting-off operation has important applications for graph connectivity problems. Shikare, Dalvi, and Dhotre [splitting-off operation for binary matroids and its applications, Graphs and Combinatorics, \(27(6) (2011), 871–882\)] extended this operation to binary matroids. In this paper, we provide a sufficient condition for preserving \(n\)-connectedness of a binary matroid under the splitting-off operation.

For positive integers \(j\) and \(k\) with \(j > k\), an \(L(j,k)\)-labelling is a generalization of classical graph coloring where adjacent vertices are assigned integers at least \(j\) apart, and vertices at distance two are assigned integers at least \(k\) apart. The span of an \(L(j,k)\)-labelling of a graph \(G\) is the difference between the maximum and minimum integers assigned to its vertices. The \(L(j,k)\)-labelling number of \(G\), denoted by \(\lambda_{j,k}(G)\), is the minimum span over all \(L(j,k)\)-labellings of \(G\). An \(m\)-\((j,k)\)-circular labelling of \(G\) is a function \(f: V(G) \to \{0,1,\ldots,m-1\}\) such that \(|f(u)-f(v)|_m \geq j\) if \(u\) and \(v\) are adjacent, and \(|f(u)-f(v)|_m \geq k\) if \(u\) and \(v\) are at distance two, where \(|x|_m = \min\{|x|,m-|x|\}\). The span of an \(m\)-\((j,k)\)-circular labelling of \(G\) is the difference between the maximum and minimum integers assigned to its vertices. The \(m\)-\((j,k)\)-circular labelling number of \(G\), denoted by \(\sigma_{j,k}(G)\), is the minimum span over all \(m\)-\((j,k)\)-circular labellings of \(G\). The \(L'(j,k)\)-labelling is a one-to-one \(L(j,k)\)-labelling, and the \(m\)-\((j,k)’\)-circular labelling is a one-to-one \(m\)-\((j,k)\)-circular labelling. Denote \(\lambda’_{j,k}(G)\) the \(L'(j,k)\)-labelling number and \(\sigma’_{j,k}(G)\) the \(m\)-\((j,k)’\)-circular labelling number. When \(j=d, k=1\), \(L(j,k)\)-labelling becomes \(L(d,1)\)-labelling. [Discrete Math. 232 (2001) 163-169] determined the relationship between \(\lambda_{2,1}(G)\) and \(\sigma_{2,1}(G)\) for a graph \(G\). We generalized the concept of path covering to the \(t\)-group path covering (Inform Process Lett (2011)) of a graph. In this paper, using group path covering, we establish relationships between \(\lambda_{4,1}(G)\) and \(\sigma_{4,1}(G)\) and between \(\lambda_{j,k}(G)\) and \(\sigma_{j,k}(G)\) for a graph \(G\) with diameter 2. Using these results, we obtain shorter proofs for the \(\sigma’_{j,k}\)-number of Cartesian products of complete graphs [J Comb Optim (2007) 14: 219-227].

We prove the following Turdn-Type result: If there are more than \(9mn/16\) edges in a simple and bipartite Eulerian digraph with vertex partition size m and n, then the graph contains a directed cycle of length \(4\) or \(6\). By using this result, we improve an upper bound for the diameter of interchange graphs.

The well known infinite families of prisms and antiprisms on the sphere were, for long time, not considered as Archimedean solids for reasons not fully understood. In this paper we describe the first two infinite families of Archimedean maps on higher genera which we call “generalized” prisms and “generalized” antiprisms.

Let \(G\) be a graph with vertex set \(V(G)\) and edge set \(E(G)\). A vertex labeling \(f: V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^*: E(G) \to \mathbb{Z}_2\) defined by \(f^*(x,y) = f(x) + f(y)\), for each edge \((x,y) \in E(G)\). For each \(i \in \mathbb{Z}_2\), let \(v_f(i) = |\{v \in V(G) : f(v) = i\}|\) and \(e_f(i) = |\{e \in E(G) : f^*(e) = i\}|\). A vertex labeling \(f\) of a graph \(G\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined as \(\{|v_f(1) – v_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). The full friendly index set of the graph \(G\), denoted by \(FFI(G)\), is defined as \(\{|e_f(1) – e_f(0)| : \text{the vertex labeling } f \text{ is friendly}\}\). In this paper, we determine \(FFI(G)\) and \(FI(G)\) for a class of cubic graphs which are twisted products of Möbius.

Let \(k\) be a non-negative integer. Two digraphs \(G = (V, A)\) and \(G’ = (V, A’)\) are \(\{k\}\)-hypomorphic if for all \(k\)-element subsets \(K\) of \(V\), the subdigraphs \(G[K]\) and \(G'[K]\) induced on \(K\) are isomorphic. The equivalence relation \(\mathcal{D}_{G,G’}\) on \(V\) is defined by: \(x \mathcal{D}_{G,G’} y\) if \(x = y\) or there exists a sequence \(x_0 = x, \ldots, x_n = y\) of elements of \(V\) satisfying \((x_i, x_{i+1}) \in A\) if and only if \((x_i, x_{i+1}) \in A’\), for all \(i\), \(0 < i k + 6\). If \(G\) and \(G’\) are two digraphs, \(\{4\}\)-hypomorphic and \(\{v – k\}\)-hypomorphic on the same vertex set \(V\) of \(uv\) vertices, and \(C\) is an equivalence class of the equivalence relation \(\mathcal{D}_{G,G’}\), then \(G'[C \setminus A]\) and \(G[C \setminus A]\) are isomorphic for all subsets \(A\) of \(V\) of at most \(k\) vertices. In particular, \(G'[C]\) and \(G[C]\) are \(\{v – k – h\}\)-hypomorphic for all \(h \in \{1, 2, \ldots, k\}\), and \(G'[C]\) and \(G[C]\) (resp. \(G’\) and \(G\)) are isomorphic. In particular, for \(k = 1\) and \(k = 4\) we obtain the result of G. Lopez and C. Rauzy [7]. As an application of the main result, we have: If \(G\) and \(G’\) are \(\{v – 4\}\)-hypomorphic on the same vertex set \(V\) of \(v > 10\) vertices, then \(G[X]\) and \(G'[X]\) are isomorphic for all subsets \(X\) of \(V\); the particular case of tournaments was obtained by Y. Boudabbous [2].