A graph G of order n is said to be borderenergetic if its energy is equal to 2n-2 and if G differs from the complete graph Kn. The first such graph was discovered in 2001, but their systematic study started only in 2015. Until now, the number of borderenergetic graphs of order n was determined for n

The graph G of order n is an L-borderenergetic which means it has the same Laplacian energy as complete Kn. In this paper, we find that combination bipartite graphs and stars can construct infinite numbers classes graphs. We give two Lborderenergetic under operators union, join their mixed. This research could provide experience for further study structural characteristics

The Laplacian and normalized Laplacian energy of G are given by expressions EL(G) = ∑n i=1 |μi − d|, EL(G) = ∑n i=1 |λi − 1|, respectively, where μi and λi are the eigenvalues of Laplacian matrix L and normalized Laplacian matrix L of G. An interesting problem in spectral graph theory is to find graphs {L,L}−noncospectral with the same E{L,L}(G). In this paper, we present graphs of order n, whi...

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...