A \(2\)-rainbow dominating function of a graph \(G\) is a function \(f\) from the vertex set \(V(G)\) to the set of all subsets of the set \(\{1,2\}\) such that for any vertex \(v \in V(G)\) with \(f(v) = \emptyset\) the condition \(\bigcup_{u \in N(v)} f(u) = \{1,2\}\) is fulfilled, where \(N(v)\) is the open neighborhood of \(v\). A rainbow dominating function \(f\) is said to be a rainbow restrained domination function if the induced subgraph of \(G\) by the vertices with label \(\emptyset\) has no isolated vertex. The weight of a rainbow restrained dominating function is the value \(w(f) = \sum_{v \in V(G)} |f(v)|\). The minimum weight of a rainbow restrained dominating function of \(G\) is called the rainbow restrained domination number of \(G\). In this paper, we continue the study of the rainbow restrained domination number. First, we classify all graphs \(G\) of order \(n\) whose rainbow restrained domination number is \(n-1\). Then, we establish an upper bound on the rainbow restrained domination number of trees.

The entire chromatic number \(\chi_c(G)\) of a plane graph \(G(V, E, F)\) is the minimum number of colors such that any two distinct adjacent or incident elements receive different colors in \(V(G) \cup E(G) \cup F(G)\). A plane graph \(G\) is called a \(1\)-tree if there exists a vertex \(u \in V(G)\) such that \(G – u\) is a forest. In this paper, it is proved that if \(G\) is a \(2\)-connected \(1\)-tree with \(\Delta(G) \geq 6\), then the entire chromatic number of \(G\) is \(\Delta(G) + 1\), where \(\Delta(G)\) is the maximum degree of \(G\).

In this paper, we compute various finite sums that alternate according to \((-1)^{\binom{n}{k}}\) involving the generalized Fibonacci and Lucas numbers for \(k = 3, 4, 5\) and even \(k\) of the form \(2^m\) with \(m \geq 1\).

In this paper, we introduce a special \((k_1A_1, k_2A_2, k_3A_3)\)-edge colouring of a graph. We shall show that for special graphs and special values of \(k_i\), \(i = 1, 2, 3\), the number of such colourings generalizes the well-known Pell numbers. Using this graph interpretation, we give a direct formula for the generalized Pell numbers. Moreover, we show some identities for these numbers.

The multiplicatively weighted Harary index (\(Hy\)-index) is a new distance-based graph invariant, which was introduced and studied by Deng et al. in [1]. For a connected graph \(G\), the multiplicatively weighted Harary index of \(G\) is defined as \(H_M(G) = \sum\limits_{\{u,v\} \subseteq V(G)} \frac{d_G(u) \cdot d_G(v)}{d_G(u,v)}\), where \(d_G(x)\) denotes the degree of vertex \(x\) and \(d_G(s,t)\) denotes the distance between vertices \(s\) and \(t\) in \(G\). In this paper, we first study a new vertex degree-based graph invariant \(M_2 – \frac{1}{2}M_1\), where \(M_1\) and \(M_2\) are ordinary Zagreb indices. We characterize the trees attaining maximum value of \(M_2 – 4M_1\) among all trees of given order. As applications, we obtain a new proof of Deng et al.’s results on trees with extremal \(H_M\)-index among all trees of given order.

In the current work, the author presents a symbolic algorithm for finding the determinant of any general nonsingular cyclic heptadiagonal matrices and the inverse of anti-cyclic heptadiagonal matrices. The algorithms are mainly based on the work presented in [A. A. Karawia, A New algorithm for inverting general cyclic heptadiagonal matrices recursively, arXiv:1011.2306v1, ICS/SCII]. The symbolic algorithms are suited for implementation using Computer Algebra Systems (CAS) such as MATLAB, MAPLE, and MATHEMATICA. An illustrative example is given.

The Wiener index of a graph is a distance-based topological index defined as the sum of distances between all pairs of vertices. In this paper, two explicit expressions for the expected value of the Wiener indices of two types of random polygonal chains are obtained.

In \([8]\), the author introduced the notion of burst errors for \(2\)-dimensional array coding systems. Also, in \([10]\), the author introduced a series of metrics called Lee-RT-Jain-Metric (LRTJ\)-metric) \([3]\) for array codes, which is a generalization of both classical Lee metric \([12]\) and array \(RT\) metric \([14]\). In this paper, we obtain sufficient conditions on the parameters of array codes equipped with \(LRTJ\)-metric for the identification and correction of burst array errors.

The concept of exterior degree of a finite group \(G\) is introduced by the author in a joint paper [13], which is the probability of randomly selecting two elements \(g\) and \(h\) in \(G\) such that \(g\wedge h = 1\). In the present paper, a necessary and sufficient condition is given for a non-cyclic group when its exterior degree achieves the upper bound \((p^2 + p – 1)/p^3\), where \(p\) is the smallest prime number dividing the order of \(G\). We also compute the exterior degree of all extra-special \(p\)-groups. Finally, for an extra-special \(p\)-group \(H\) and a group \(G\) where \(G/Z^\wedge(G)\) is a \(p\)-group, we will show that \(d^\wedge(G) = d^\wedge(H)\) if and only if \(G/Z^\wedge(G) \cong H/Z^\wedge(H)\), provided that \(d^\wedge(G) \neq 11/32\).

Let \(G\) be a unicyclic graph on \(n \geq 3\) vertices. Let \(A(G)\) be the adjacency matrix of \(G\). The eigenvalues of \(A(G)\) are denoted by \(\lambda_1(G) \geq \lambda_2(G) \geq \cdots \geq \lambda_n(G)\), which are called the eigenvalues of \(G\). Let the unicyclic graphs \(G\) on \(n\) vertices be ordered by their least eigenvalues \(\lambda_n(G)\) in non-decreasing order. For \(n \geq 14\), the first six graphs in this order are determined.