This paper introduces a method for finding closed forms for certain sums involving squares of binomial coefficients. We use this method to present an alternative approach to a problem of evaluating a different type of sums containing squares of the numbers from
Catalan’s triangle.

In this paper, we deal with a special kind of hypergraph decomposition. We show that there exists a decomposition of the 3-uniform hypergraph \(\lambda K_v^{(3)}\) into a special kind of hypergraph \(K_{4}^{(3)} – e\) whose leave has at most two edges, for any positive integers \(v \geq 4 \) and \(\lambda\).

For a connected graph \(G\) of order \(n \geq 2\) and a linear ordering \(s = v_1, v_2, \ldots, v_n\) of \(V(G)\), define \(d(s) = \sum_{i=1}^{n-1} d(v_i, v_{i+1})\), where \(d(v_i, v_{i+1})\) is the distance between \(v_i\) and \(v_{i+1}\). The traceable number \(t(G)\) and upper traceable number \(t^+(G)\) of \(G\) are defined by \(t(G) = \min\{d(s)\}\) and \(t^+(G) = \max\{d(s)\}\), respectively, where the minimum and maximum are taken over all linear orderings \(s\) of \(V(G)\). The traceable number \(t(v)\) of a vertex \(v\) in \(G\) is defined by \(t(v) = \min\{d(s)\}\), where the minimum is taken over all linear orderings \(s\) of \(V(G)\) whose first term is \(v\). The \({maximum\; traceable \;number}\) \(t^*(G)\) of \(G\) is then defined by \(t^*(G) = \max\{t(v) : v \in V(G)\}\). Therefore, \(t(G) \leq t^*(G) \leq t^+(G)\) for every nontrivial connected graph \(G\). We show that \(t^*(G) \leq \lfloor \frac{t(G)+t^+(G)+1}{2}\rfloor\) for every nontrivial connected graph \(G\) and that this bound is sharp. Furthermore, it is shown that for positive integers \(a\) and \(b\), there exists a nontrivial connected graph \(G\) with \(t(G) = a\) and \(t^*(G) = b\) if and only if \(a \leq b \leq \left\lfloor \frac{3n}{2} \right\rfloor\).

Let \(G\) be a simple graph with \(n\) vertices and \(m\) edges, and let \(\lambda_1\) and \(\lambda_2\) denote the largest and second largest eigenvalues of \(G\). For a nontrivial bipartite graph \(G\), we prove that:
(i) \(\lambda_1 \leq \sqrt{m – \frac{3-\sqrt{5}}{2}}\), where equality holds if and only if \(G \cong P_4\);
(ii) If \(G \ncong P_n\), then \(\lambda_1 \leq \sqrt{{m} – (\frac{5-\sqrt{17}}{2})}\), where equality holds if and only if \(G \cong K_{3,3} – e\);
(iii) If \(G\) is connected, then \(\lambda_2 \leq \sqrt{{m} – 4{\cos}^2(\frac{\pi}{n+1})}\), where equality holds if and only if \(G \cong P_{n,2} \leq n \leq 5\);
(iv) \(\lambda_2 \geq \frac{\sqrt{5}-1}{2}\), where equality holds if and only if \(G \cong P_4\);
(v) If \(G\) is connected and \(G \ncong P_n\), then \(\lambda_2 \geq \frac{5-\sqrt{17}}{2}\), where equality holds if and only if \(G \cong K_{3,3} – e\).

Let \(n\) be a positive integer. Denote by \(PG(n,q)\) the \(n\)-dimensional projective space over the finite field \(\mathbb{F}_q\) of order \(q\). A blocking set in \(PG(n,q)\) is a set of points that has non-empty intersection with every hyperplane of \(PG(n,q)\). A blocking set is called minimal if none of its proper subsets are blocking sets. In this note, we prove that if \(PG(n_i,q)\) contains a minimal blocking set of size \(k_i\) for \(i \in \{1,2\}\), then \(PG(n_1 + n_2 + 1,q)\) contains a minimal blocking set of size \(k_1 + k_2 – 1\). This result is proved by a result on groups with maximal irredundant covers.

A graph is said to be edge-transitive if its automorphism group acts transitively on its edge set. In this paper, all connected cubic edge-transitive graphs of order \(12p\) or \(12p^2\) are classified.

For any \(n\geq 7\), we prove that there exists a tournament of order \(n\), such that for each pair of distinct vertices there exists a path of length \(2\).

A \((k, t)\)-list assignment \(L\) of a graph \(G\) assigns a list of \(k\) colors available at each vertex \(v\) in \(G\) and \(|\bigcup_{v\in V(G)}L(v)| = t\). An \(L\)-coloring is a proper coloring \(c\) such that \(c(v) \in L(v)\) for each \(v \in V(G)\). A graph \(G\) is \((k,t)\)-choosable if \(G\) has an \(L\)-coloring for every \((k, t)\)-list assignment \(L\).
Erdős, Rubin, and Taylor proved that a graph is \((2, t)\)-choosable for any \(t > 2\) if and only if a graph does not contain some certain subgraphs. Chareonpanitseri, Punnim, and Uiyyasathian proved that an \(n\)-vertex graph is \((2,t)\)-choosable for \(2n – 6 \leq t \leq 2n – 4\) if and only if it is triangle-free. Furthermore, they proved that a triangle-free graph with \(n\) vertices is \((2, 2n – 7)\)-choosable if and only if it does not contain \(K_{3,3} – e\) where \(e\) is an edge. Nakprasit and Ruksasakchai proved that an \(n\)-vertex graph \(G\) that does not contain \(C_5 \vee K_{n-2}\) and \(K_{4,4}\) for \(k \geq 3\) is \((k, kn – k^2 – 2k)\)-choosable. For a non-2-choosable graph \(G\), we find the minimum \(t_1 \geq 2\) and the maximum \(t_2\) such that the graph \(G\) is not \((2, t_i)\)-choosable for \(i = 1, 2\) in terms of certain subgraphs. The results can be applied to characterize \((2, t)\)-choosable graphs for any \(t\).

Let \(G\) be the circuit graph of any connected matroid. It is proved that the circuit graph of a connected matroid with at least three circuits is \(E_2\)-Hamiltonian.

The Randić index \(R(G)\) of a graph \(G\) is defined by \(R(G) = \sum\limits_{uv} \frac{1}{\sqrt{d(u)d(v)}}\), where \(d(u)\) is the degree of a vertex \(u\) in \(G\) and the summation extends over all edges \(uv\) of \(G\). In this work, we give sharp lower bounds of \(R(G) + g(G)\) and \(R(G) . g(G)\) among \(n\)-vertex connected triangle-free graphs with Randić index \(R\) and girth \(g\).