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 \(\chi’_{avd}(C_5) = \Delta(C_5) + 3\).

Let \(G\) be a simple graph of order \(n\). A dominating set of \(G\) is a set \(S\) of vertices of \(G\) such that every vertex of \(G\) is either in \(S\) or adjacent to a vertex in \(S\). The domination polynomial of \(G\) is defined as \(D(G, x) = \sum_{i=0}^{n} d(G, i)x^i\), where \(d(G, i)\) denotes the number of dominating sets of \(G\) of size \(i\). In this paper, we demonstrate that cycles are uniquely determined by their domination polynomials.

The third-order Randić index of a graph \(G\) is defined as \(R_s(G) = \sum_{u_1u_2u_3u_4} \frac{1}{\sqrt{d(u_1) d(u_2) d(u_3) d(u_4)}}\), where the summation is taken over all possible paths of length three in \(G\). In this paper, we first derive a recursive formula for computing the third-order Randić index of a double hexagonal chain. Furthermore, we establish upper and lower bounds for the third-order Randić index and characterize the double hexagonal chains that achieve the extremal third-order Randić index.

The decycling index of a digraph \(D\) is defined to be the minimum number of arcs in a set whose removal from \(D\) leaves an acyclic digraph. In this paper, we obtain some results on the decycling index of bipartite tournaments.

In this paper two authentication codes with multiple arbiters are constructed to protect the communication system against the attacks from the opponent, transmitter, receiver and dishonest arbiters. The first construction takes advantage of set theory to give an authentication codes with two arbiters that resists collusion attacks from dishonest arbiters and participators availably. The second construction makes full use of of Reed- Solomon-code (\(RS\)-code) and \((k, n)\)-threshold scheme to give an authentication codes with \(n\) arbiters that effectively prevents multiple arbiters from cheating.

A directed Toeplitz graph is a digraph with a Toeplitz adjacency matrix. In this paper we contribute to [6]. The paper [6] investigates the hamiltonicity of the directed Toeplitz graphs \(T_n\langle s_1,s_2,…, s_k;t_1, t_2,…,t_l\rangle\) with \(s_2 = 2\) and in particular those with \(s_3 = 3\). In this paper we extend this investigation to \(s_2 = 3\) with \(s_1 =t_1 =1\).