A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the...

The Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph are the characteristic polynomials of its Laplacian matrix, signless Laplacian matrix and normalized Laplacian matrix, respectively. In this paper, we mainly derive six reduction procedures on the Laplacian, signless Laplacian and normalized Laplacian characteristic polynomials of a graph which can be used to construct larger Laplacian, signless Laplacian and normalized Laplacian cospectral graphs, respectively....

Let $A\left(G\right)$ be the adjacency matrix of $G$. The characteristic polynomial of the adjacency matrix $A$ is called the characteristic polynomial of the graph $G$ and is denoted by $\phi (G,\lambda )$ or simply $\phi \left(G\right)$. The spectrum of $G$ consists of the roots (together with their multiplicities) ${\lambda }_1\left(G\right)\ge {\lambda }_2\left(G\right)\ge ...\ge {\lambda }_n\left(G\right)$ of the equation $\phi (G,\lambda )=0$. The largest root ${\lambda }_1\left(G\right)$ is referred to as the spectral radius of $G$. A $‡$-shape is a tree with exactly two of its vertices having maximal degree 4. We will denote by $G(l_1,l_2,...,l_7)$$...$