# 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

## Keywords:

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

## Abstract

Let G be an undirected simple graph on n vertices and sigma2(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>=3 is an odd number then alpha(G)=2 and G is a disconnected graph; for n>=4 is an even number then 2=<alpha(G)<=(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<=k<=(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)  