In \(1982\), Beutelspacher and Brestovansky determined the \(2\)-color Rado number of the equation \[x_1+ x_2 x + \ldots +x_{m-1} =x_{ m} \] for all \(m \geq 3\). Here we extend their result by determining the 2-color Rado number of the equation \[x_1 +x_2 + \dots + x_n = y_1 +y_2+ \ldots + y_k\] for all \(n \geq 2\) and \(k \geq 2\). As a consequence, we determine the 2-color Rado number of \[x_1+ x_2 + \ldots + x_n = a_1 y_1 + \dots + a_\ell y_\ell\] in all cases where \(n \geq 2\) and \(n \geq a_1 + \dots + a_\ell\), and in most cases where \(n \geq 2\) and \(2n \geq a_1 + \dots + a_\ell\).

The subdivision graph \(S(G)\) of a graph \(G\) is the graph obtained by inserting a new vertex into every edge of \(G\). The set of inserted vertices of \(S(G)\) is denoted by \(I(G)\). Let \(G_1\) and \(G_2\) be two vertex-disjoint graphs. The subdivision-edge-vertex join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(V(G_2)\). The subdivision-edge-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \ominus G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(I(G_1)\) to every vertex in \(I(G_2)\). The subdivision-vertex-edge join of \(G_1\) and \(G_2\), denoted by \(G_1 \odot G_2\), is the graph obtained from \(S(G_1)\) and \(S(G_2)\) by joining every vertex in \(V(G_1)\) to every vertex in \(I(G_2)\). In this paper, we obtain the formulas for resistance distance of \(G_1 \odot G_2\), \(G_1 \ominus G_2\), and \(G_1 \odot G_2\).

A hypergraph is intersecting if any two different edges have exactly one common vertex, and an \(n\)-quasicluster is an intersecting hypergraph with \(n\) edges, each one containing at most \(n\) vertices, and every vertex is contained in at least two edges. The Erdős-Faber-Lovász Conjecture states that the chromatic number of any \(n\)-quasicluster is at most \(n\). In the present note, we prove the correctness of the conjecture for a new infinite class of \(n\)-quasiclusters using a specific edge coloring of the complete graph.

Let \(G\) be a graph of order \(n\) and let \(\Phi(G, \lambda) = \det(\lambda I_n – L(G)) = \sum_{k=0}^{n}(-1)^k c_k(G) \lambda^{n-k}\) be the characteristic polynomial of the Laplacian matrix of a graph \(G\). In this paper, we identify the minimal Laplacian coefficients of unicyclic graphs with \(n\) vertices and diameter \(d\). Finally, we characterize the graphs with the smallest and the second smallest Laplacian-like energy among the unicyclic graphs with \(n\) vertices and fixed diameter \(d\).

The balanced hypercube, which is a variant of the hypercube, is proposed as a novel inter-processor network. Among the attractive properties of the balanced hypercube, the most special one is that each processor has a backup processor sharing the same neighborhood. A connected graph \(G\) with at least \(2m + 2\) vertices is said to be \(m\)-extendable if it possesses a matching of size \(m\) and every such matching can be extended to a perfect matching of \(G\). In this paper, we prove that the balanced hypercube \(BH_n\) is \(m\)-extendable for every \(m\) with \(1 \leq m \leq 2n – 2\), and our result is optimal.

A set \(W \subseteq V(G)\) is called a resolving set, if for each two distinct vertices \(u, v \in V(G)\) there exists \(w \in W\) such that \(d(u, w) \neq d(v, w)\), where \(d(x, y)\) is the distance between the vertices \(x\) and \(y\). A resolving set for \(G\) with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a unique basis graph. In this paper, we study some properties of unique basis graphs.

In this paper, we study the number of 1-factors and edge-colorings of the Möbius ladder graphs. We find exact formulae for such numbers and show that there are exponentially many 1-factors and edge-colorings in such graphs. As applications, we show that every “man-made” triangular embedding for \(K_{12m+7}\), by combining the current graphs with those of Youngs and Ringel, permits exponentially many “Grünbaum colorings” (i.e., 3-edge-colored triangulations in such a way that each triangle receives three distinct colors).

Multi-receiver authentication codes with dynamic sender (\(DMRA\)-codes) are extensions of traditional group communication systems in which any member of a group can broadcast an authenticated message such that all other group members can individually verify its authenticity, and some malicious participants of the group cannot successfully impersonate the potential sender, or substitute a transmitted message. In this paper, a construction of \(DMRA\)-code will be given using linear code and its unconditional security is also guaranteed.

We consider the problem of finding quasiperiodicities in Fibonacci strings. A factor \(u\) of a string \(y\) is a cover of \(y\) if every letter of \(y\) falls within some occurrence of \(u\) in \(y\). A string \(v\) is a seed of \(y\) if it is a cover of a superstring of \(y\). A left seed of a string \(y\) is a prefix of \(y\) that is a cover of a superstring of \(y\). Similarly, a right seed of a string \(y\) is a suffix of \(y\) that is a cover of a superstring of \(y\). In this paper, we present some interesting results regarding quasiperiodicities in Fibonacci strings; we identify all covers, left/right seeds, and seeds of a Fibonacci string and all covers of a circular Fibonacci string.

We investigate a modifications of the well-known irregularity strength of graphs, namely the total edge irregularity strength and the total vertex irregularity strength. In this paper, we determine the exact value of the total edge (vertex) irregularity strength for convex polytope graphs with pendent edges.