The aim of this paper is to answer a question proposed by Li \([2]\) and prove that no connected bi-normal Cayley graph other than cycles of even length is \(3\)-arc-transitive.

Using new ways to label edges in an ordered tree, this paper introduces two bijections between bicoloured ordered trees and non-crossing partitions. Consequently, enumeration results of non-crossing partitions specified with several parameters are derived.

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.