Related papers: Parameters of Quotient-Polynomial Graphs

As a generalization of orbit-polynomial and distance-regular graphs, we introduce the concept of a quotient-polynomial graph. In these graphs every vertex $u$ induces the same regular partition around $u$, where all vertices of each cell…

In this paper we characterize "large" regular graphs using certain entries in the projection matrices onto the eigenspaces of the graph. As a corollary of this result, we show that "large" association schemes become $P$-polynomial…

Let $\Gamma$ denote an undirected, connected, regular graph with vertex set $X$, adjacency matrix $A$, and ${d+1}$ distinct eigenvalues. Let ${\mathcal A}={\mathcal A}(\Gamma)$ denote the subalgebra of Mat$_X({\mathbb C})$ generated by $A$.…

We show an inequality involving the third largest or second smallest dual eigenvalues of $Q$-polynomial association schemes of class at least three. Also we characterize dual-tight $Q$-polynomial association schemes of class three. Our…

One may think of a $d$-class association scheme as a $(d+1)$-dimensional matrix algebra over $\mathbb{R}$ closed under Schur products. In this context, an imprimitive scheme is one which admits a subalgebra of block matrices, also closed…

A graph $X$ is said to be a pattern polynomial graph if its adjacency algebra is a coherent algebra. In this study we will find a necessary and sufficient condition for a graph to be a pattern polynomial graph. Some of the properties of the…

We classify the Q-polynomial association schemes with $m_{1} = 4$ which are partially metric with respect to the nearest neighbourhood relation. An association scheme is partially metric with respect to a relation $R_1$ if the scheme graph…

We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when…

In this article we determine feasible parameter sets for (what could potentially be) commutative association schemes with noncyclotomic eigenvalues that are of smallest possible rank and order. A feasible parameter set for a commutative…

Symmetric edge polytopes, also called adjacency polytopes, are lattice polytopes determined by simple undirected graphs. We introduce the integer array \(\mathrm{maxf}(n,m)\) giving the maximum number of facets of a symmetric edge polytope…

We present the tables of feasible parameters of primitive $3$-class $Q$-polynomial association schemes and $4$- and $5$-class $Q$-bipartite association schemes (on up to $2800$, $10000$, and $50000$ vertices, respectively), accompanied by a…

The adjacency-diametrical matrix (AD matrix) of a connected graph $G$ with diameter $d$, denoted by $AD(G)$, is the matrix indexed by the vertices of $G$ in which the $(i,j)$-entry of $AD(G)$ is $1$ if $d_G(v_i,v_j)=1$, is $d$ if…

Semialgebraic graphs are graphs whose vertices are points in $\mathbb{R}^d$, and adjacency between two vertices is determined by the truth value of a semialgebraic predicate of constant complexity. We show how to harness polynomial…

Let $G$ be a connected graph on $n$ vertices with adjacency matrix $A_G$. Associated to $G$ is a polynomial $d_G(x_1,\dots, x_n)$ of degree $n$ in $n$ variables, obtained as the determinant of the matrix $M_G(x_1,\dots,x_n)$, where…

We study functional graphs generated by quadratic polynomials over prime fields. We introduce efficient algorithms for methodical computations and provide the values of various direct and cumulative statistical parameters of interest. These…

We prove a necessary and sufficient condition for a symmetric association scheme to be a Q-polynomial scheme.

Let ${\cal M}$ denote the Bose--Mesner algebra of a commutative $d$-class association scheme ${\mathfrak X}$ (not necessarily symmetric), and $\Gamma$ denote a (strongly) connected (directed) graph with adjacency matrix $A$. Under the…

Matrices are the most common representations of graphs. They are also used for the representation of algebras and cluster algebras. This paper shows some properties of matrices in order to facilitate the understanding and locating…

Preferential attachment lies at the heart of many network models aiming to replicate features of real world networks. To simulate the attachment process, conduct statistical tests, or obtain input data for benchmarks, efficient algorithms…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…