A path in an edge-colored graph \(G\), where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a \(k\)-connected graph \(G\) and an integer \(k\) with \(1 \leq k \leq \kappa\), the rainbow \(k\)-connectivity \(rc_k(G)\) of \(G\) is defined as the minimum integer \(j\) for which there exists a \(j\)-edge-coloring of \(G\) such that any two distinct vertices of \(G\) are connected by \(k\) internally disjoint rainbow paths. Denote by \(K_{r,r}\) an \(r\)-regular complete bipartite graph. Chartrand et al. in in “G. Chartrand, G.L. Johns, K.A.McKeon, P. Zhang, The rainbow connectivity of a graph, Networks \(54(2009), 75-81”\) left an open question of determining an integer \(g(k)\) for which the rainbow \(k\)-connectivity of \(K_{r,r}\) is \(3\) for every integer \(r \geq g(k)\). This short note is to solve this question by showing that \(rc_k(K_{r,r}) = 3\) for every integer \(r \geq 2k\lceil\frac{k}{2}\rceil\), where \(k \geq 2\) is a positive integer.

Let \(G\) be a connected graph with edge set \(E(G)\). The Balaban index of \(G\) is defined as \(J(G) = \frac{m}{\mu+1} \sum_{uv \in E(G)} ({D_uD_v})^{-\frac{1}{2}}\) where \(m = |E(G)|\), and \(\mu\) is the cyclomatic number of \(G\), \(D_u\) is the sum of distances between vertex \(u\) and all other vertices of \(G\). We determine \(n\)-vertex trees with the first several largest and smallest Balaban indices.

For a graph \(G = (V, E)\), \(X \subseteq V\) is a global dominating set if \(X\) dominates both \(G\) and the complement graph \(\bar{G}\). A set \(X \subseteq V\) is a packing if its pairwise members are distance at least \(3\) apart. The minimum number of vertices in any global dominating set is \(\gamma_g(G)\), and the maximum number in any packing is \(\rho(G)\). We establish relationships between these and other graphical invariants, and characterize graphs for which \(\rho(G) = \rho(\bar{G})\). Except for the two self-complementary graphs on \(5\) vertices and when \(G\) or \(\bar{G}\) has isolated vertices, we show \(\gamma_g(G) \leq \lfloor n/2 \rfloor\), where \(n = |V|\).

The inverse degree \(r(G)\) of a finite graph \(G = (V, E)\) is defined by \(r(G) = \sum_{v\in V} \frac{1}{deg(v)}\) where \(deg(v)\) is the degree of \(v\) in \(G\). Erdős \(et\) \(al\). proved that, if \(G\) is a connected graph of order \(n\), then the diameter of \(G\) is less than \((6r(G) + \sigma(1))\frac{\log n}{\log \log n}\). Dankelmann et al. improved this bound by a factor of approximately \(2\). We give the sharp upper bounds for trees and unicyclic graphs, which improves the above upper bounds.

Let \(\gamma_c(G)\) be the connected domination number of \(G\) and \(\gamma_{tr}(G)\) be the tree domination number of \(G\). In this paper, we study the generalized Petersen graphs \(P(n,k)\), prove \(\gamma_c(P(n, k)) = \gamma_{tr}(P(n, k))\) and show their exact values for \(k = 1, 2, \ldots, \lfloor n/2 \rfloor\).

Given a parity-check matrix \({H}\) with \(n\) columns, an \(\ell\)-subset \(T\) of \(\{1,2,\ldots,n\}\) is called a stopping set of size \(\ell\) for \({H}\) if the \(\ell\)-column submatrix of \({H}\) consisting of columns with coordinate indexes in \(T\) has no row of Hamming weight one. The size of the smallest non-empty stopping sets for \({H}\) is called the stopping distance of \({H}\).

In this paper, the stopping distance of \({H}_{m}(2t+1)\), parity-check matrices representing binary \(t\)-error-correcting \(BCH\) codes, is addressed. It is shown that if \(m\) is even then the stopping distance of this matrix is three. We conjecture that this property holds for all integers \(m \geq 3\).

For the sequence satisfying the recurrence relation of the second order, we establish a general summation theorem on the infinite series of the reciprocal product of its two consecutive terms. As examples, several infinite series identities are obtained on Fibonacci and Lucas numbers, hyperbolic sine and cosine functions, as well as the solutions of Pell equation.

The directed \(\overrightarrow{P}_k\)-graph of a digraph \(D\) is obtained by representing the directed paths on \(k\) vertices of \(D\) by vertices. Two such vertices are joined by an arc whenever the corresponding directed paths in \(D\) form a directed path on \(k+1\) vertices or a directed cycle on \(k\) vertices in \(D\). In this paper, we give a necessary and sufficient condition for two digraphs with isomorphic \(\overrightarrow{P}_3\)-graphs. This improves a previous result, where some additional conditions were imposed.

In this paper, we study quaternary quasi-cyclic \((QC)\) codes with even length components. We determine the structure of one generator quaternary \(QC\) codes whose cyclic components have even length. By making use of their structure, we establish the size of these codes and give a lower bound for minimum distance. We present some examples of codes from this family whose Gray images have the same Hamming distances as the Hamming distances of the best known binary linear codes with the given parameters. In addition, we obtain a quaternary \(QC\) code that leads to a new binary non-linear code that has parameters \((96, 2^{26}, 28)\).

Let \(G\) be a simple graph, and let \(p\) be a positive integer. A subset \(D \subseteq V(G)\) is a \(p\)-dominating set of the graph \(G\), if every vertex \(v \in V(G) – D\) is adjacent to at least \(p\) vertices in \(D\). The \(p\)-domination number \(\gamma_p(G)\) is the minimum cardinality among the \(p\)-dominating sets of \(G\). A subset \(I \subseteq V(G)\) is an independent dominating set of \(G\) if no two vertices in \(I\) are adjacent and if \(I\) is a dominating set in \(G\). The minimum cardinality of an independent dominating set of \(G\) is called independence domination number \(i(G)\).

In this paper, we show that every block-cactus graph \(G\) satisfies the inequality \(\gamma_2(G) \geq i(G)\) and if \(G\) has a block different from the cycle \(C_3\), then \(\gamma_2(G) \geq i(G) + 1\). In addition, we characterize all block-cactus graphs \(G\) with \(\gamma_2(G) = i(G)\) and all trees \(T\) with \(\gamma_2(T) = i(T) + 1\).