Two graphs are defined to be adjointly equivalent if their complements are chromatically equivalent. In \([2, 7]\), Liu and Dong et al. give the first four coefficients \(b_0\), \(b_1\), \(b_2\), \(b_3\) of the adjoint polynomial and two invariants \(R_1\), \(R_2\), which are useful in determining the chromaticity of graphs. In this paper, we give the expression of the fifth coefficient \(b_4\), which brings about a new invariant \(R_3\). Using these new tools and the properties of the adjoint polynomials, we determine the chromatic equivalence class of \(\overline{B_{n-9,1,5}}\).

Given a tournament \(T = (V, A)\), a subset \(X\) of \(V\) is an interval of \(T\) provided that for any \(a, b \in X\) and \(x \in V – X\), \((a, x) \in A\) if and only if \((b, x) \in A\). For example, \(\emptyset\), \(\{x\}\) (\(x \in V\)), and \(V\) are intervals of \(T\), called trivial intervals. A tournament whose intervals are trivial is indecomposable; otherwise, it is decomposable. With each indecomposable tournament \(T\), we associate its indecomposability graph \(\mathbb{I}(T)\) defined as follows: the vertices of \(\mathbb{I}(T)\) are those of \(T\) and its edges are the unordered pairs of distinct vertices \(\{x, y\}\) such that \(T -\{x, y\}\) is indecomposable. We characterize the indecomposable tournaments \(T\) whose \(\mathbb{I}(T)\) admits a vertex cover of size \(2\).

Let \(G\) be a simple graph. A harmonious coloring of \(G\) is a proper vertex coloring such that each pair of colors appears together on at most one edge. The harmonious chromatic number \(h(G)\) is the least number of colors in such a coloring. In this paper, it is shown that if \(T\) is a tree of order \(n\) and \(\Delta(T) \geq \frac{n}{2}\), then \(h(T) = \Delta(T) + 1\), where \(\Delta(T)\) denotes the maximum degree of \(T\). Let \(T_1\) and \(T_2\) be two trees of order \(n_1\) and \(n_2\), respectively, and \(F = T_1 \cup T_2\). In this paper, it is shown that if \(\Delta(T_i) = \Delta_i\) and \(\Delta_i \geq \frac{n_i}{2}\), for \(i = 1, 2\), then \(h(F) \leq \Delta(F) + 2\). Moreover, if \(\Delta_1 = \Delta_2 = \Delta \geq \frac{n_i}{2}\), for \(i = 1, 2\), then \(h(F) = \Delta + 2\).

Hamming graph \(H(n, k)\) has as vertex set all words of length \(n\) with symbols taken from a set of \(k\) elements. Suppose \(L\) denotes the set \(\bigcup_{i=0}^{n+1}\Omega_l\) with \(\Omega_l=\{\sum\limits_{i\in I_1}e_i^1+\sum\limits_{i\in I_2}e_i^2+\ldots+\sum\limits_{i\in I_k}e_i^k|I_j\cap I_j’=\emptyset (j\neq j’),|\bigcup_{j=1}^kI_j|=l\}\) for \(0\leq l\leq n\) and \(\Omega_{n+1}\). For any two elements \(x, y \in L\), define \(x \leq y\) if and only if \(y = I\) or \(I^x_j \leq I^y_j\) for some \(1 \leq j \leq k\). Then \(L\) is a lattice, denoted by \(L_o\). Reversing the above partial order, we obtain the dual of \(L_o\), denoted by \(L_r\). This article discusses their geometric properties and computes their characteristic polynomials.

The paper considers two-dimensional linear codes with sub-block structure in RT-spaces \([2-5,7]\) whose error location techniques are described in terms of various sub-blocks. Upper and lower-bounds are given for the number of check digits required with any error locating code in RT-spaces.

Let \(k \geq 0\) be an integer. Oblong (pronic) numbers are numbers of the form \(O_k = k(k+1)\). In this work, we set a new integer sequence \(B = B_n(k)\) defined as \(B_0 = 0\), \(B_1 = 1\), and \(B_n = O_k B_{n-1} – B_{n-2}\) for \(n \geq 2\), and then derive some algebraic relations on it. Later, we give some new results on balancing numbers via oblong numbers.

This note deals with the computation of the factorization number \(F_2(G)\) of a finite group \(G\). By using the Möbius inversion formula, explicit expressions of \(F_2(G)\) are obtained for two classes of finite abelian groups, improving the results of “Factorization numbers of some finite groups”, Glasgow Math. J. (2012).

Given a set of vertices \(S = \{v_1, v_2, \ldots, v_k\}\) of a connected graph \(G\), the metric representation of a vertex \(v\) of \(G\) with respect to \(S\) is the vector \(r(v|S) = (d(v, v_1), d(v, v_2), \ldots, d(v, v_k))\), where \(d(v, v_i)\), \(i \in \{1, \ldots, k\}\), denotes the distance between \(v\) and \(v_i\). \(S\) is a resolving set of \(G\) if for every pair of distinct vertices \(u, v\) of \(G\), \(r(u|S) \neq r(v|S)\). The metric dimension \(\dim(G)\) of \(G\) is the minimum cardinality of any resolving set of \(G\). Given an ordered partition \(\Pi = \{P_1, P_2, \ldots, P_t\}\) of vertices of a connected graph \(G\), the partition representation of a vertex \(v\) of \(G\), with respect to the partition \(\Pi\), is the vector \(r(v|\Pi) = (d(v, P_1), d(v, P_2), \ldots, d(v, P_t))\), where \(d(v, P_i)\), \(1 \leq i \leq t\), represents the distance between the vertex \(v\) and the set \(P_i\), that is \(d(v, P_i) = \min_{u \in P_i} \{d(v, u)\}\). \(\Pi\) is a resolving partition for \(G\) if for every pair of distinct vertices \(u, v\) of \(G\), \(r(u|\Pi) \neq r(v|\Pi)\). The partition dimension \(\mathrm{pd}(G)\) of \(G\) is the minimum number of sets in any resolving partition for \(G\). Let \(G\) and \(H\) be two graphs of order \(n\) and \(m\), respectively. The corona product \(G \odot H\) is defined as the graph obtained from \(G\) and \(H\) by taking one copy of \(G\) and \(n\) copies of \(H\) and then joining, by an edge, all the vertices from the \(i\)-th copy of \(H\) with the \(i\)-th vertex of \(G\). Here, we study the relationship between \(\mathrm{pd}(G \odot H)\) and several parameters of the graphs \(G \odot H\), \(G\), and \(H\), including \(\dim(G \odot H)\), \(\mathrm{pd}(G)\), and \(\mathrm{pd}(H)\).

We study: combination and permutation graphs. We introduce some familes to be: combination graphs and permutation graphs.

An \(L(2, 1)\)-labeling of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all nonnegative integers such that \(|f(x) – f(y)| \geq 2\) if \(d(x, y) = 1\) and \(|f(x) – f(y)| \geq 1\) if \(d(x, y) = 2\), where \(d(x, y)\) denotes the distance between \(x\) and \(y\) in \(G\). The \(L(2, 1)\)-labeling number, \(\lambda(G)\), of \(G\) is the smallest number \(k\) such that \(G\) has an \(L(2, 1)\)-labeling \(f\) with \(\max\{f(v) : v \in V(G)\} = k\). In this paper, we present a new characterization on \(d\)-disk graphs for \(d > 1\). As an application, we give upper bounds on the \(L(2, 1)\)-labeling number for these classes of graphs.