On the sum of the second Zagreb index and its reciprocal index for bicyclic graphs

The hyperbolic Sombor index (HSO) is a recently introduced vertex-degree–based topological index that originates from the geometric properties of a hyperbola. In this work, we explore several mathematical properties of the HSO index, as well as revisit and refine some previously reported results. We first provide a counterexample to the claim that HSO(G) always increases with the addition of an edge and establish a sufficient condition under which this monotonicity holds. We then present refined versions of some existing results and proofs. Furthermore, we establish sharp upper and lower bounds for the HSO index across various classes of graphs, including trees, unicyclic graphs, and bicyclic graphs, and characterize the corresponding extremal graphs that attain these bounds. Finally, we identify the first eight minimal trees, as well as seven minimal unicyclic and bicyclic graphs with respect to HSO.

The geometric quadratic (GQ) index is a recently introduced degree-based topological descriptor, and Kumar et al. observed that it is potentially a very good molecular descriptor. In this paper, we characterize the extremal graphs (chemical) and trees concerning the geometric quadratic index of a given order and size. Then, we determine the -vertex trees, unicyclic and bicyclic graphs with the maximum, the second, the third, the fourth, the fifth, and the sixth maximum geometric quadratic indices.

Bicyclic Inverses of Bicyclic Graphs with a Unique Perfect Matching

On b-Coloring of Unicyclic and Bicyclic Graphs

Wiener index has been extensively studied for several decades because of its applications in chemistry. Many variants of Wiener index were defined and their corresponding bounds were explored. In this work, we introduced the concept of geodetic-Wiener index by considering the number of geodesics between any pair of vertices. We used the concept of projection of a vertex to a subgraph to decompose the structure into subtrees. Simple spirocyclic graphs are bicyclic graphs whose cycles share a common vertex. Using the idea of partial Wiener index, we determined the bounds of geodetic-Wiener index with respect to other distance-based topological indices for simple spirocyclic graphs.

The field related to indices was developed by researchers for various purposes. Optimization is one of the purposes used by researchers in different situations. In this article, a generalized Sombor index is considered. This work is related to the idea of optimization in the families of bicyclic graphs, trees, and unicyclic graphs. We investigated optimal values in the stated families by means of well-known transformations. The transformations include the following: Transformation A, Transformation B, Transformation C, and Transformation D. Transformation A and Transformation B increase the value of the generalized Sombor index, while Transformation C and Transformation D are used for minimal values.

The Sombor matrix $S(G)$ of a graph $G$ was introduced by Gutman in 2021, in which the $(i, j)$-entry is equal to $\sqrt{deg^{2}(u_{i})+deg^{2}(u_{j})}$ if the vertices $u_{i}$ and $u_{j}$ are adjacent in $G$, and zero otherwise, where $deg(u_{i})$ denotes the degree of vertex $u_{i}$ in $G$. In this paper, we obtain the extremal graphs with the first two maximum Sombor spectral radii in bicyclic graphs.

Mostar index of certain classes of bicyclic graphs

The graph in this research is a simple and connected graph with as vertex set and as an edge set. We used deductive axiomatic and pattern recognition method. Local irregularity vertex coloring is defined as mapping as vertex irregular -labeling and where . The conditions for to be a local irregularity vertex coloring, if with as irregularity vertex labeling and for every . The minimum number of colors produced from local irregularity vertex coloring of graph is called chromatic number local irregularity, denoted by . In this research, we analyze about the local irregularity vertex coloring and determine the chromatic number of local irregularity of bicyclic graphs.

An outer independent double Roman dominating function (OIDRDF) of a graph \( G \) is a function \( f:V(G)\rightarrow\{0,1,2,3\} \) satisfying the following conditions: (i) every vertex \( v \) with \( f(v)=0 \) is adjacent to a vertex assigned 3 or at least two vertices assigned 2; (ii) every vertex \( v \) with \( f(v)=1 \) has a neighbor assigned 2 or 3; (iii) no two vertices assigned 0 are adjacent. The weight of an OIDRDF is the sum of its function values over all vertices, and the outer independent double Roman domination number \( \gamma_{oidR}(G) \) is the minimum weight of an OIDRDF on \( G \). Ahangar et al. [Appl. Math. Comput. 364 (2020) 124617] established that for every tree \( T \) of order \( n \geq 4 \), \( \gamma_{oidR}(T)\leq\frac{5}{4}n \) and posed the question of whether this bound holds for all connected graphs. In this paper, we show that for a unicyclic graph \( G \) of order \( n \), \( \gamma_{oidR}(G) \leq \frac{5n+2}{4} \), and for a bicyclic graph, \( \gamma_{oidR}(G) \leq \frac{5n+4}{4} \). We further characterize the graphs attaining these bounds, providing a negative answer to the question posed by Ahangar et al.

An adjacent vertex distinguishing edge coloring of graphs is a proper edge coloring of graphs such that no two adjacent vertices have the same set of colors, and the minimum number of the colors required for such coloring is called adjacent vertex distinguishing edge chromatic number. In this paper, we characterize the relationships of adjacent vertex distinguishing edge chromatic numbers between all bicyclic graphs as well as some tricyclic base graphs and their subgraphs, respectively. By the way, the open problem proposed by Zhang et al. [Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15(5) (2002) 623–626] is answered completely on a bicyclic graph.

The topological index (TI), sometimes referred to as the connectivity index, is a molecular descriptor calculated based on the molecular graph of a chemical compound. Topological indices (TIs) are numeric parameters of a graph used to characterize its topology and are usually graph-invariant. The generalized power-sum connectivity index (GPSCI) for the graph is ΩYα(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)+dΩ(ϱ)dΩ(ϱ))α, while the second form of the GPSCI is defined as Yβ(Ω)=∑ζϱ∈E(Ω)(dΩ(ζ)dΩ(ζ)×dΩ(ϱ)dΩ(ϱ))β. In this paper, we investigate Yβ in the family of trees, unicyclic graphs, and bicyclic graphs. We determine optimal graphs in the desired families for Yβ using certain mappings. For graphs with maximal values, two mappings are used, namely A and B, while for graphs with minimal values, mapping C and mapping D are considered.

Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: eABC=eABC(Υ)=∑vivj∈E(Υ)edi+dj−2didj, where di is the degree of the vertex vi in Υ. In this paper, we prove that the double star DSn−3,1 is the second maximal graph with respect to the eABC index of trees of order n. We give an upper bound on eABC of unicyclic graphs of order n and characterize the maximal graphs. The graph K1∨(P3∪(n−4)K1) gives the maximal graph with respect to the eABC index of bicyclic graphs of order n. We present several relations between eABC(Υ) and ABC(Υ) of graph Υ. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.

Maximum energy bicyclic graphs containing two odd cycles with one common vertex

Let HG (x, y) be the expected hitting time from vertex x to vertex y for the first time on a simple connected graph G and ?(G) = max,y?V(G)xG H (x, y). Lettn G be the set of simple connected graphs with n vertices and t pendant vertices. In this paper, we proved the upper bound of the ?(G) for G ?tn G and determined the extremal graph in all n-vertex bicyclic graphs with given t pendant vertices.

Topological indices are widely used to analyze and predict the physicochemical properties of compounds, and have good application prospects. Recently, the Euler Sombor index was introduced, which is defined as \begin{document}$ \begin{align} EP(G) = \sum\limits_{v_iv_j\in E(G)}\sqrt{d(v_i)^2+d(v_j)^2+d(v_i)d(v_j)}.& \end{align} $\end{document} As the latest index with geometry motivation, it has excellent discrimination and predictive ability for compounds, in addition to mathematical practicality. The unicyclic graphs and bicyclic graphs are composed of various chemical structures, and are of particular importance in the study of topological indices. In this paper, the maximal and minimal values of Euler Sombor index for all unicyclic and bicyclic graphs are determined, and the corresponding extremal graphs are characterized.

Extremal spectral radius of degree-based weighted adjacency matrices of graphs with given order and size

Proof of a Conjecture on the Atom-Bond Sum-Connectivity Index of Bicyclic Graphs

For a connected graph \(G\), the edge Mostar index \(Mo_e(G)\) is defined as \(Mo_e(G)=\sum\limits_{e=uv \in E(G)}|m_u(e|G) - m_v(e|G)|\), where \(m_u(e|G)\) and \(m_v(e|G)\) are respectively, the number of edges of \(G\) lying closer to vertex \(u\) than to vertex \(v\) and the number of edges of \(G\) lying closer to vertex \(v\) than to vertex \(u\). We determine a sharp upper bound for the edge Mostar index on bicyclic graphs and identify the graphs that achieve the bound, which disproves a conjecture proposed by Liu et al. [Iranian J. Math. Chem. 11(2) (2020) 95--106].