Sharp bounds for the general Randić index of graphs with fixed number of vertices and cyclomatic number

The cyclomatic number, denoted by $\gamma$, of a graph $G$ is the minimum number of edges of $G$ whose removal makes $G$ acyclic. Let $\mathscr{G}_{n}^{\gamma}$ be the class of all connected graphs with order $n$ and cyclomatic number $\gamma$. In this paper, we characterized the graphs in $\mathscr{G}_{n}^{\gamma}$ with minimum general Randić index for $\gamma\geq 3$ and $1\leq\alpha\leq \frac{39}{25}$. These extend the main result proved by A. Ali, K. C. Das and S. Akhter in 2022. The elements of $\mathscr{G}_{n}^{\gamma}$ with maximum general Randić index were also completely determined for $\gamma\geq 3$ and $\alpha\geq 1$

Sufficient Conditions for a Graph to Be ℓ-Connected, ℓ-Deficient, ℓ-Hamiltonian and ℓ−-Independent in Terms of the Forgotten Topological Index

The forgotten topological index of a (molecule) graph is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. It is meaningful and difficult to explore sufficient conditions for a given graph keeping certain properties in graph theory. In this paper, we mainly explore sufficient conditions in terms of the forgotten topological index for a graph to be ℓ-connected, ℓ-deficient, ℓ-Hamiltonian and ℓ−-independent, respectively. The conditions cannot be dropped

A Complete Characterization of Bipartite Graphs with Given Diameter in Terms of the Inverse Sum Indeg Index

In 2010, Vukičević introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range of applications. Let Bnd be the class of bipartite graphs of order n and diameter d. In this paper, we mainly characterize the bipartite graphs in Bnd with the maximal inverse sum indeg index. Bipartite graphs with the largest, second-largest, and smallest inverse sum indeg indexes are also completely characterized

The Nordhaus–Gaddum-type inequalities for the Zagreb index and co-index of graphs

AbstractLet k≥2 be an integer, a k-decomposition (G1,G2,…,Gk) of the complete graph Kn is a partition of its edge set to form k spanning subgraphs G1,G2,…,Gk. In this contribution, we investigate the Nordhaus–Gaddum-type inequality of a k-decomposition of Kn for the general Zagreb index and a 2-decomposition for the Zagreb co-indices, respectively. The corresponding extremal graphs are characterized

A Complete Characterization of Bidegreed Split Graphs with Four Distinct α-Eigenvalues

It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. A graph is d-extremal (resp. dα-extremal) if it has diameter d and exactly d+1 distinct eigenvalues (resp. α-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either d˜ or d^, then we say it is (d˜,d^)-bidegreed. In this paper, we present a complete classification of the connected bidegreed 3α-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively

A Complete Characterization of Bidegreed Split Graphs with Four Distinct <i>α</i>-Eigenvalues

It is a well-known fact that a graph of diameter d has at least d+1 eigenvalues. A graph is d-extremal (resp. dα-extremal) if it has diameter d and exactly d+1 distinct eigenvalues (resp. α-eigenvalues), and a graph is split if its vertex set can be partitioned into a clique and a stable set. Such graphs have a diameter of at most three. If all vertex degrees in a split graph are either d˜ or d^, then we say it is (d˜,d^)-bidegreed. In this paper, we present a complete classification of the connected bidegreed 3α-extremal split graphs using the association of split graphs with combinatorial designs. This result is a natural generalization of Theorem 4.6 proved by Goldberg et al. and Proposition 3.8 proved by Song et al., respectively