Let \(G\) be a finite abelian group. The critical number \(cr(G)\) of \(G\) is the least positive integer \(m\) such that every subset \(A \subseteq G \setminus \{0\}\) of cardinality at least \(m\) spans \(G\), i.e., every element of \(G\) can be expressed as a nonempty sum of distinct elements of \(A\). Although the exact values of \(cr(G)\) have been recently determined for all finite abelian groups, the structure of subsets of cardinality \(cr(G) – 1\) that fail to span \(G\) remains characterized except when \(|G|\) is even or \(|G| = pq\) with \(p, q\) primes. In this paper, we characterize these extremal subsets for \(|G| \geq 36\) and \(|G|\) even, or \(|G| = pq\) with \(p, q\) primes and \(q \geq 2p + 3\).

In this paper, we give a criterion to judge whether a linear code over the ring is self-dual. Moreover, we introduce the generating set in standard form for the cyclic codes over \(F_p + vF_p\), and characterize the structure of cyclic codes over the ring. Then we prove that cyclic codes over the ring are principally generated and obtain the unique generating idempotent for cyclic codes of length \(n\), where \(n\) is coprime to \(p\).

Let \(G\) be a finite abelian group, and let \(S\) be a sequence over \(G\). For a sequence \(S\), denote by \(f(S)\) the number of elements in \(G\) that can be expressed as the sum of a nonempty subsequence of \(S\). In this paper, we determine all sequences \(S\) that contain no zero-sum subsequences and satisfy \(f(S) \leq 2|S| – 1\).

For given a graph \(H\), agraphic sequence \(\pi = (d_1, d_2,\ldots, d_n)\) is said to be potentially \(H\)-graphic if there exists a realization of \(m\) containing \(H\) asa subgraph. Let \(K_m- H\) be the graph obtained from \(K_m\), by removing the edges set \(E(H)\) where \(H\) is a subgraph of \(K_m\). In this paper, we characterize potentially \(K_{2,5}\)-graphic sequences. This characterization implies a special case of a theorem due to Yin \(et \;al. [26]\).

The harmonic index of a graph \(G\) is defined as the sum of weights Tay raey of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) are the degrees of the vertices \(u\) and \(v\) in \(G\), respectively. In this paper, we give a sharp lower bound on the harmonic index of trees with a perfect matching in terms of the number of vertices. A sharp lower bound on the harmonic index of trees with a given size of matching is also obtained.

Graphs \(S[n,k]\) are introduced as the graphs obtained from the Sierpiński graphs \(S(n, k)\) by contracting edges that lie in no complete subgraph \(K_k\). The family \(S[n,k]\) generalizes the previously studied class of Sierpiński gasket graphs \(S_k\). We investigate various properties of graphs \(S[n,k]\), particularly focusing on hamiltonicity and chromatic number.

Let \(G\) be a graph. The Randić index of \(G\) is the sum of the weights \((d(u)d(v))^{-\frac{1}{2}}\) of all edges \(uv\) of \(G\), where \(d(u)\) and \(d(v)\) denote the degrees of vertices \(u\) and \(v\) in \(G\). In this paper, we establish a sharp upper bound for the Randić index \(R(G)\) among all unicyclic graphs \(G\) with \(n\) vertices, \(k\) pendant vertices, and \(n \geq 3k\), where \(k \geq 3\).

Let \(G\) be a simple quadrangulation on a closed surface \(F^2\). Two reductions for quadrangulations are defined in this paper: face-contraction and \(4\)-cycle removal. We define four types of irreducibility:

1. \(G\) is \({irreducible}\) if any face-contraction breaks the simplicity of \(G\).
2. \(G\) is \(\mathcal{D}_3\)-\({irreducible}\) if \(G\) has minimum degree at least 3 and any face-contraction or 4-cycle removal breaks simplicity or reduces minimum degree to less than 3.
3. \(G\) is \(\mathcal{K}_3\)\({-irreducible}\) if \(G\) is 3-connected and any face-contraction or 4-cycle removal breaks simplicity or 3-connectedness.
4. \(G\) is \(\mathcal{S}_4\) \({-irreducible}\) if \(G\) has no separating 4-cycle and any face-contraction breaks simplicity or creates a separating 4-cycle.

In [7] that, except for the sphere and projective plane, irreducibility and \(\mathcal{D}_3\)-irreducibility of quadrangulations are equivalent. In this paper, we prove that for all surfaces, \(\mathcal{D}_3\)-irreducibility and \(\mathcal{K}_3\)-irreducibility are equivalent. Additionally, we prove that for the sphere, projective plane, and torus, \(\mathcal{D}_3\)-irreducibility and \(\mathcal{S}_4\)-irreducibility are equivalent, but this does not hold for surfaces of high genus.

An adjacent vertex-distinguishing edge coloring ,avd-coloring for short, of a graph \(G\) is a proper edge coloring of \(G\) such that no pair of adjacent vertices are incident to the same set of colors. We denote the avd-chromatic number of \(G\) by \(\chi’_{avd}(G)\), which is the smallest integer \(k\) such that \(G\) has an avd-coloring with \(k\) colors, and the maximum degree of \(G\) by \(\Delta(G)\). In this paper, we prove that \(\chi’_{avd}(G) \leq \Delta(G) + 4\) for every planar graph \(G\) without isolated edges whose girth is at least five. Notably, this bound is nearly sharp, as \(\chi’_{avd}(C_5) = \Delta(C_5) + 3\).