In this paper, we give necessary and sufficient conditions for a real symmetric matrix and, in particular, the distance $D(H_n)$ of helm graph $H_n$ to have their Moore-Penrose inverses as sum Laplacian-like rank-one matrix. As consequence, present short proof inverse formula, given by Goel (Linear Algebra Appl. 621:86-104, 2021), when $n$ is even. Further, derive formula singular that analogou...

a watching system in a graph $g=(v, e)$ is a set $w={omega_{1}, omega_{2}, cdots, omega_{k}}$, where $omega_{i}=(v_{i}, z_{i}), v_{i}in v$ and $z_{i}$ is a subset of closed neighborhood of $v_{i}$ such that the sets $l_{w}(v)={omega_{i}: vin omega_{i}}$ are non-empty and distinct, for any $vin v$. in this paper, we study the watching systems of line graph $k_{n}$ which is called triangular grap...

Let $G$ be a connected graph, and let $D[G]$ denote the double graph of $G$. In this paper, we first derive closed-form formulas for different distance based topological indices for $D[G]$ in terms of that of $G$. Finally, as illustration examples, for several special kind of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for some distance based topologica...

BACKGROUND Hypoepiluminescence microscopy (HELM) is a new dermoscopic approach for analysis of pigmented skin lesions (PSLs) after surgical excision. OBJECTIVES The objective was to verify whether this method could provide additional morphologic information for diagnostic or didactic purposes compared to conventional epiluminescence microscopy (ELM). PATIENTS AND METHODS Thirty-one PSLs fro...

Let n ? 4 . The helm graph H on 2 ? 1 vertices is obtained from the wheel W by adjoining a pendant edge to each vertex of outer cycle Suppose even. D : = [ d i j ] be distance matrix In this paper, we first show that det ( ) 3 Next, find L and vector u such + ? We also prove an interlacing property between eigenvalues