Abstract A graph G is $$\alpha $$ α -excellent if every vertex of contained in some maximum independent set . In this paper, we characterize bipartite graphs, unicyclic simplicial chordal block and show that generalized Petersen -excellent.

Recently, a novel degree-based molecular structure descriptor, called Sombor index was introduced. Let G=(V(G),E(G)) be graph. Then, the of G is defined as SO(G)=?uv?E(G)dG2(u)+dG2(v). In this paper, we give some lemmas that can used to compare indices between two graphs. With these lemmas, determine graph with maximum SO among all cacti n vertices and k cut edges. Furthermore, unique p pendant...

Let G be a connected graph on n vertices and D(G) its distance matrix. The formula for computing the determinant of this matrix in terms number is known when either tree or unicyclic graph. In work we generalize these results, obtaining any whose block decomposition consists edges, bicyclic graphs.

Let $$\Gamma =(K_{n},H^-)$$ be a signed complete graph whose negative edges induce subgraph H. The index of $$\Gamma$$ is the largest eigenvalue its adjacency matrix. In this paper, we study when H unicyclic graph. We show that among all graphs order $$n>5$$ k and maximizes index, triangle with remaining vertices being pendant at same vertex triangle.