# THE STRUCTURE OF GRAPHS ON $$n$$ VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO $$n-2$$

## Authors

• Do Nhu An Nha Trang University, Viet Nam

## Keywords:

Connected graph, Disconnected graph, Maximum independent set, Regular graph.

## Abstract

Let $$G$$ be an undirected simple graph on n vertices and $$\sigma^2(G)=n-2$$ (degree sum of any two non-adjacent vertices in $$G$$ is equal to $$n-2$$) and $$\alpha(G)$$ be the cardinality of an maximum independent set of $$G$$. In $$G$$, a vertex of degree $$(n-1)$$ is called total vertex. We show that, for $$n\geq3$$ is an odd number then $$\alpha(G)=2$$ and $$G$$ is a disconnected graph; for $$n\geq4$$ is an even number then $$2\leq\alpha(G)\leq(n+2)/2$$, where if $$\alpha(G)=2$$ then $$G$$ is a disconnected graph, otherwise $$G$$ is a connected graph, $$G$$ contains  $$k$$ total vertices and $$n-k$$ vertices of degree $$\delta=(n-2)/2$$, where $$0\leq k\leq(n-2)/2$$. In particular, when $$k=0$$ then $$G$$ is an $$(n-2)/2$$-Regular graph.

## References

An, D. N. (2008). Some problems about Hamiltonian cycle in special graphs. The 2nd International Conference on Theories and Applications of Computer Science (ICTACS’09) - Journal of Science and Technology, Vietnam Academy of Science and Technology, 6(5A-Special issue), 57-66.

An, D. N. (2019). Recognizing the Hamiltonian graph on n vertices with σ_2 (G)=n-1 is an easy problem. International Journal of Advanced Research in Computer Science, 10(2), 42-45.

Graham, R. L., Grötschel, M., & Lovász, L. (1995). Handbook of combinatorics (1st Ed.). North Holland.

04-10-2021

## Section

Natural Sciences and Technology

## How to Cite

An, D. N. (2021). THE STRUCTURE OF GRAPHS ON $$n$$ VERTICES WITH THE DEGREE SUM OF ANY TWO NONADJACENT VERTICES EQUAL TO $$n-2$$. Dalat University Journal of Science, 11(4), 55-62. https://doi.org/10.37569/DalatUniversity.11.4.830(2021)