Anthony J. Macula constructed a \(d\)-disjunct matrix \(\delta(n,d,k)\) in \([1]\), and we now know it is determined by one type of pooling space. In this paper, we give some properties of \(\delta(n,d,k)\) and its complement \(\delta^c(n,d,k)\).

Let \(S\) be a primitive non-powerful signed digraph. The base \(l(S)\) of \(S\) is the smallest positive integer \(l\) such that for all ordered pairs of vertices \(i\) and \(j\) (not necessarily distinct), there exists a pair of \(SSSD\) walks of length \(t\) from \(i\) to \(j\) for each integer \(t \geq l\). In this work, we use \(PNSSD\) to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(n)\) be the largest value of \(l(S)\) for \(S \in\) \(PNSSD\), and \(L(n) = \{l(S) | S \in PNSSD\}\). For \(n \geq 3\), we show \(L(n) = \{2, 3, \ldots, 2n\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop whose bases attain \(l(n)\).

For a graph \(G = (V, E)\) and a binary labeling \(f : V(G) \to \mathbb{Z}_2\), let \(v_f(i) = |f^{-1}(1)|\). The labeling \(f\) is said to be friendly if \(|v_f(1) – v_f(0)| \leq 1\). Any vertex labeling \(f : V(G) \to \mathbb{Z}_2\) induces an edge labeling \(f^* : E(G) \to \mathbb{Z}_2\) defined by \(f^*(xy) =| f(x) – f(y)|\). Let \(e_f(i) = |f^{*-1}(i)|\). The friendly index set of the graph \(G\), denoted by \(FI(G)\), is defined by

\[FI(G) = \{|e_f(1) – e_f(0)| : f \text{ is a friendly vertex labeling of } G\}.\]

In \([15]\) Lee and Ng conjectured that the friendly index sets of trees will form an arithmetic progression. This conjecture has been mentioned in \([17]\) and other manuscripts. In this paper, we will first determine the friendly index sets of certain caterpillars of diameter four. Then we will disprove the conjecture by presenting an infinite number of trees whose friendly index sets do not form an arithmetic progression.

Let \(S\) be a primitive non-powerful signed digraph of order \(n\). The base of a vertex \(u\), denoted by \(l_S(u)\), is the smallest positive integer \(l\) such that there is a pair of SSSD walks of length \(i\) from \(u\) to each vertex \(v \in V(S)\) for any integer \(t \geq l\). We choose to order the vertices of \(S\) in such a way that \(l_S(1) \leq l_S(2) \leq \ldots \leq l_S(n)\), and call \(l_S(k)\) the \(k\)th local base of \(S\) for \(1 \leq k \leq n\). In this work, we use PNSSD to denote the class of all primitive non-powerful signed symmetric digraphs of order \(n\) with at least one loop. Let \(l(k)\) be the largest value of \(l_S(k)\) for \(S \in\) PNSSD, and \(L(k) = \{l_S(k) | S \in PNSSD\}\). For \(n \geq 3\) and \(1 \leq k \leq n-1\), we show \(I(k) = 2n – 1\) and \(L(k) = \{2, 3, \ldots, 2n-1\}\). Further, we characterize all primitive non-powerful signed symmetric digraphs whose \(k\)th local bases attain \(I(k)\).

Let \(\mathcal{U}_n(k)\) denote the set of all unicyclic graphs on \(n\) vertices with \(k\) (\(k \geq 1\)) pendant vertices. Let \(\diamondsuit_4^k\) be the graph on \(n\) vertices obtained from \(C_4\) by attaching \(k\) paths of almost equal lengths at the same vertex. In this paper, we prove that \(\diamondsuit_4^k\) is the unique graph with the largest Laplacian spectral radius among all the graphs in \(\mathcal{U}_n(k)\), when \(n \geq k + 4\).

For graphs \(G_1, G_2, \ldots, G_m\), the Ramsey number \(R(G_1, G_2, \ldots, G_m)\) is defined to be the smallest integer \(n\) such that any \(m\)-coloring of the edges of the complete graph \(K_n\) must include a monochromatic \(G_i\) in color \(i\), for some \(i\). In this note, we establish several lower and upper bounds for some Ramsey numbers involving quadrilateral \(C_4\), including:\(R(C_4, K_9) \leq 32,
19 \leq R(C_4, C_4, K_4)\leq 22, 31 \leq R(C_4, C_4, C_4, K_4) \leq 50, 52 \leq R(C_4, K_4, K_4) \leq 72, 42 \leq R(C_4, C_4, K_3, K_5) \leq 76, 87\leq R(C_4, C_4, K_4, K_4) \leq 179.\)

We consider the problem of determining the \(Q\)-integral graphs, i.e., the graphs with integral signless Laplacian spectrum. First, we determine some infinite series of such graphs having the other two spectra (the usual one and the Laplacian) integral. We also completely determine all \((2, s)\)-semiregular bipartite graphs with integral signless Laplacian spectrum. Finally, we give some results concerning \((3, 4)\) and \((3, 5)\)-semiregular bipartite graphs with the same property.

Let \(G\) be a connected graph. For any two vertices \(u\) and \(v\), let \(d(u, v)\) denote the distance between \(u\) and \(v\) in \(G\). The maximum distance between any pair of vertices is called the diameter of \(G\) and denoted by \(diam(G)\). A radio-labeling (or multi-level distance labeling) with span \(k\) for \(G\) is a function \(f\) that assigns to each vertex a label from the set \(\{0, 1, 2, \ldots, k\}\) such that the following holds for any vertices \(u\) and \(v\): \(|f(u) – f(v)| \geq diam(G) – d(u, v) + 1\). The radio number of \(G\) is the minimum span over all radio-labelings of \(G\). The square of \(G\) is a graph constructed from \(G\) by adding edges between vertices of distance two apart in \(G\). In this article, we completely determine the radio number for the square of any path.

Let \(G\) be a simple connected graph containing a perfect matching. \(G\) is said to be BM-extendable if every matching \(M\) whose induced subgraph is a bipartite graph extends to a perfect matching of \(G\). In this paper, for recognizing BM-extendable graphs, we present some conditions in terms of vertex degrees, including the degree sum conditions, the minimum degree conditions, and the Fan-type condition. Furthermore, we show that all these conditions are best possible in some sense.

Let \(u\) be an odd vertex of a bipartite graph \(B\) and suppose that \(f : V(B) \to \mathbb{N}\) is a function such that \(f(u) = \left\lceil d_B(u)/2 \right\rceil\) and \(f(v) = \left\lceil d_B(v)/2 \right\rceil + 1\) for \(v \in V(B) \setminus u\), where \(d_B(v)\) is the degree of \(v\) in \(B\). In this paper, we prove that \(B\) is \(f\)-choosable.