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 \(g\).

Hammack and Livesay introduced a new graph operation \(G^{(k)}\) for a graph \(G\), which they called the \(k\)th inner power of \(G\). A graph \(G\) is Hamiltonian if it contains a spanning cycle. In this paper, we show that \(C^{(k)}_n(n \geq 3, k \geq 2)\) is Hamiltonian if and only if \(n\) is odd and \(k = 2\), where \(C_n\) is the cycle with \(n\) vertices.

Let \(a(v)\) and \(g(v)\) denote the least possible area and the least possible number of lattice points in the interior of a convex lattice \(v\)-gon, respectively. Many lower and upper bounds for \(a(v)\) and \(g(v)\) are known for every \(v\). However, the exact values of these two functions are only known for \(v \leq 10\) and \(v \in \{12, 13, 14, 16, 18, 20, 22\}\). The purpose of this paper is to answer the following Open Question 1 from \([13]\): What is the exact value of \(a(11)\)? We answer this question by proving that \(a(11) = 21.5\). On our way to achieve this goal, we also prove that \(g(11) = 17\).

The edge-face total chromatic number of \(3\)-regular Halin graphs was shown to be \(4\) or \(5\) in \([5]\). In this paper, we shall provide a necessary and sufficient condition to characterize \(3\)-regular Halin graphs with edge-face total chromatic number equal to four.