The first and second multiplicative Zagreb indices of a simple graph \(G\) are defined as:
\[ \prod_1(G) = \prod_{u \in V(G)} d_G(u)^2
\text{and}
\prod_2(G) = \prod_{uv \in E(G)} d_G(u)d_G(v),\]
where \(d_G(u)\) denotes the degree of the vertex \(u\) of \(G\). In this paper, we establish strict lower bounds on the first and second multiplicative Zagreb indices of various graph operations in terms of the first and second multiplicative Zagreb indices and multiplicative sum Zagreb index of their components.

This paper contributes to the study of automorphism groups of \(2-(v, k, 1)\) designs. Let \(\mathcal{D}\) be a \(2-(v, 31, 1)\) design and \(G \leq Aut(\mathcal{D})\) be block-transitive and point-primitive. If \(G\) is unsolvable, then \(Soc(G)\), the socle of \(G\), is not isomorphic to \(^2F_4(q)\).

The Randić index of a graph \(G\), denoted by \(R(G)\), is defined as the sum of \(\frac{1}{d(u)d(v)}\) over all edges \(uv\) of \(G\), where \(d(u)\) denotes the degree of a vertex \(u\) in \(G\). Denote by \(\nu(G)\) the matching number, i.e., the number of edges in a maximum matching of \(G\). A conjecture of AutoGraphiX on the relation between the Randić index and the matching number of a connected graph \(G\) states: for any connected graph of order \(n \geq 3\) with Randić index \(R(G)\) and matching number \(\mu(G)\),
\[ R(G) – \mu(G) \leq \sqrt{\lfloor\frac{n+4}{7}\rfloor \lfloor \frac{6n+2}{7} \rfloor} -\lfloor \frac{n+4}{7}\rfloor \]
with equality if and only if \(G\) is a complete bipartite graph \(K_{p,q}\) with \(p = \mu(G) = \left\lfloor \frac{n+4}{2} \right\rfloor\), which was proposed by Aouchiche et al. In this paper, we confirm this conjecture for some classes of graphs.

A 2-semiarc is a pointset \(\mathcal{S}_2\) with the property that the number of tangent lines to \(\mathcal{S}_2\) at each of its points is two. Using theoretical results and computer-aided search, we provide the complete classification of 2-semiarcs in \(PG(2, q)\) for \(q \leq 7\), determine the spectrum of their sizes for \(q \leq 9\), and prove existence results for \(q = 11\) and \(q = 13\). Additionally, for several sizes of 2-semiarcs in \(PG(2, q)\) with \(q \leq 7\), classification results have been obtained through theoretical proofs.

In this paper, we concentrate on rooted general maps on all surfaces(orientable and nonorientable) without regard to genus and present the enumerating equation with respect to vertices and edges, which is a Riccati’s equation. To solve it, a new solution in continued fraction form is given. As two especial cases, the corresponding results of rooted general maps and rooted monopole maps on all surfaces with respect to edges regardless of genus are obtained.

For a tree \(T\), the set of leaves of \(T\) is denoted by \(Leaf(T)\), and the subtree \(T – Leaf(T)\) is called the \({stem} of T\). We prove that if a connected graph \(G\) either satisfies \(\sigma_{k+1}(G) \geq |G| – k – 1\) or has no vertex set of size \(k+1\) such that the distance between any two of its vertices is at least \(4\), then \(G\) has a spanning tree whose stem has at most \(k\) leaves, where \(\sigma_{k+1}(G)\) denotes the minimum degree sum of \(k+1\) independent vertices of \(G\). Moreover, we show that the condition on \(\sigma_{k+1}(G)\) is sharp. Additionally, we provide another similar sufficient degree condition for a claw-free graph to have such a spanning tree.

We prove that every connected subcubic graph G has two spanning trees \(T_1,T_2\) such that every component of \(G – E(T_1)\) is a path of length at most \(3\), and every component of \(G – E(T_2)\) is either a path of length at most \(2\) or a cycle.

A graph \(X\) is said to be \({End-completely-regular}\) (\({End-inverse}\)) if its endomorphism monoid \(End(X)\) is completely regular (inverse). In this paper, we demonstrate that if \(X + Y\) is End-completely-regular, then both \(X\) and \(Y\) are End-completely-regular. We present several approaches to construct new End-completely-regular graphs via the join of two graphs with specific conditions. Notably, we determine the End-completely-regular joins of bipartite graphs. Furthermore, we prove that \(X + Y\) is End-inverse if and only if \(X + Y\) is End-regular and both \(X\) and \(Y\) are End-inverse. Additionally, we determine the End-inverse joins of bipartite graphs.

The tensor product of two graphs \(G_1\) and \(G_2\), denoted by \(G_1 \times G_2\), is defined as the graph with vertex set \(\{(x, y): x \in V(G_1), y \in V(G_2)\}\) and edge set \(\{(x_1, y_1)(x_2, y_2): x_1x_2 \in E(G_1), y_1y_2 \in E(G_2)\}\). Very recently, Zhang, Zheng, and Mamut showed that if \(\delta(G_1) \geq 2\) and \(G_2\) does not belong to a well-characterized class \(\mathcal{G}\) of graphs, then \(G_1 \times G_2\) admits a nowhere-zero \(3\)-flow. However, it remains unclear whether \(G_1 \times G_2\) admits a nowhere-zero \(3\)-flow if \(\delta(G_1) \geq 2\) and \(G_2\) belongs to \(\mathcal{G}\), especially for the simplest case \(G_2 = K_2\). The main objective of this paper is to show that for any graph \(G\) with \(2 \leq \delta(G) \leq \Delta(G) \leq 3\), \(G \times K_2\) admits a nowhere-zero \(3\)-flow if and only if either every cycle in \(G\) contains an even number of vertices of degree \(2\) or every cycle in \(G\) contains an even number of vertices of degree \(3\). We also extend the sufficiency of this result to graphs \(G \times K_2\), where all odd vertices in \(G\) are of degree \(3\).

The notion of \(SDVFA\) (Strong Deterministic Variable Finite Automaton) of order \((s,t)\) was previously introduced by the author \([12]\). In this paper, we demonstrate the equivalence of \(SDVFA\) of order \((s,t)\) with DFA (Deterministic Finite Automaton), \(VDPA\) (Variable Deterministic Pushdown Automaton), NFA (Nondeterministic Finite Automaton), and \(\epsilon\)-NFA (extended Nondeterministic Finite Automaton). This equivalence is established by presenting conversions between \(SDVFA\) and \(DFA, VDFA, NFA\) (\(\epsilon\)-NFA), and vice versa.