A Hadamard matrix is balanced splittable if some subset of its rows has the property that dot product every two distinct columns takes at most values. This definition was introduced by Kharaghani and Suda in 2019, although equivalent formulations have been previously studied using different terminology. We collate previous results phrased terms matrices, real flat equiangular tight frames, sphe...

The Mostar index of a graph was defined by Do\v{s}li\'{c}, Martinjak, \v{S}krekovski, Tipuri\'{c} Spu\v{z}evi\'{c} and Zubac in the context study properties chemical graphs. It measures how far given is from being distance-balanced. In this paper, we determine two well-known families graphs: Fibonacci cubes Lucas cubes.

A graph is balanced if its clique-vertex incidence matrix is balanced, i.e., it does not contain a square submatrix of odd order with exactly two ones per row and per column. Interval graphs, obtained as intersection graphs of intervals of a line, are well-known examples of balanced graphs. A circular-arc graph is the intersection graph of a family of arcs on a circle. Circular-arc graphs gener...

We introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let G=(V,E) be a connected graph with a weight (supply/demand) function p:V→ {−1,+1} satisfying p(V )= ∑ j∈V p(j)=0. The objective is to partition G into (V1, V2) such that G[V1] and G[V2] are connected, |p(V1)|, |p(V2)|≤cp, and max{ |V1| |V2| , |V2| |V1|}≤cs, for some constants cp and cs. When G is 2-connec...

We construct twelve infinite families of pseudocyclic and non-amorphic association schemes, in which each nontrivial relation is a strongly regular graph. Three of the twelve families generalize the counterexamples to A. V. Ivanov’s conjecture by Ikuta and Munemasa [13].

A block negacyclic Bush-type Hadamard matrix of order 36 is used in a symmetric BGW(26, 25, 24) with zero diagonal over a cyclic group of order 12 to construct a twin strongly regular graph with parameters v=936, k=375, l=m=150 whose points can be partitioned in 26 cocliques of size 36. The same Hadamard matrix then is used in a symmetric BGW(50, 49, 48) with zero diagonal over a cyclic group o...