We use Hadamard's determinantal inequality and its generalization to prove some upper bounds on the energy of a graph in terms degrees, average 2-degrees number common neighbors vertices. Also, we an relating one arbitrary subgraph it.

We show that the set of vertices in a treeT of smallest weight balance is the (branchweight) centroid. We characterize the set of edges in T with smallest weight-edge difference in terms of the vertices of smallest weight balance. A similar characterization is obtained for the set of edges in T with smallest distance-edge difference in terms of the vertices of smallest distance balance. This yi...

astigmatism’s defect has not yet been formulated in physics and medical physics books theoretically. in this paper, we are trying to formulate the astigmatism’s defect. the results of formulation show that “the degree of astigmatism is negative reverse the distance that vision’s condition of astigmatism’s chart lines change conditions in dimness angle from dimness condition at infinite distance...

A digraph such that for each its vertex, vertices of the outneighbourhood have different in-degrees and vertices of the inneighbourhood have different out-degrees, will be called an HI-digraph. In this paper, we give a characterization of sequences of pairs of outand in-degrees of HI-digraphs.

A coloring of a graph G is an assignment of colors to the vertices of G in such a way that every pair of adjacent vertices receive different colors. If G admits a coloring using r colors, G is said to be r-colorable, Graph coloring has a number of applications. In many of them, we encounter a situation in which several vertices of G are already colored. We are required to find a coloring which ...

We study the problem of computing shortest paths in so-called dense distance graphs. Every planar graph G on n vertices can be partitioned into a set of O(n/r) edge-disjoint regions (called an r-division) with O(r) vertices each, such that each region has O( √ r) vertices (called boundary vertices) in common with other regions. A dense distance graph of a region is a complete graph containing a...

An almost self-centered graph is a connected of order $n$ with exactly $n-2$ central vertices, and an peripheral $n-1$ vertices. We determine (1) the maximum girth $n$; (2) independence number radius $r$; (3) minimum $k$-regular graph; (4) size (5) possible degrees (6) vertices degree in $n-4$ which second largest possible. Whenever extremal graphs have neat form, we also describe them.

Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we introduce some general transformations that increase the terminal distance spectral radius of a connected graph and characterize the extremal trees with respect to the terminal distance spectral radius among all trees with a fixed number of pendant ver...