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Summary

Description
English: The Chang Graphs.

The Chang Graphs are strongly regular graphs with parameters (28,12,6,4). On the right the tree Chang Graphs; these graphs are generated by selecting a proper switching set. On the left the originating [Triangular Graph]s T8:

the vertices in the switching set are green, the deleted edges are red and new added ones are phantom blue.
Date
Source Own work
Author Claudio Rocchini
Permission
(Reusing this file)		 CC-BY 3.0

References

Many thanks to Nadia Hamoudi for paper "The Chang graphs" at Nadia Hamoudi "The Chang graphs" archive copy at the Wayback Machine

A note: nauty software shows an automorphism group really poor for this graphs.

Source

Dirty C++ source code of graph generation and display: 

/*********************************
 * Drawing the Chang Graphs
 * (C) 2010 Claudio Rocchini
 * CC-BY 3.0
 * Many thanks to Nadia Hamoudi for
 * "The Chang graphs".
 *********************************/
#include <stdio.h>
#include <math.h>
#include <conio.h>
#include <vector>
#include <set>
#include <algorithm>
const double PI = 3.1415926535897932384626433832795;

class point2 { public: double x,y; };

typedef std::pair<size_t,size_t> edge;

class graph {
public:
	size_t nv;
	std::vector<edge> edges;
	int find_edge( const edge & e ) const {
		std::vector<edge>::const_iterator i = std::find(edges.begin(),edges.end(),e);
		return  i==edges.end() ? -1 : i - edges.begin();
	}
};

bool is_strong_regular( const graph & g, int & K, int & LAMDA, int & MU ) {
	int i,j,k;
	std::vector<bool> MA(g.nv*g.nv); std::fill(MA.begin(),MA.end(),false);
	std::vector<edge>::const_iterator q;
	K = -1;
	for(q=g.edges.begin();q!=g.edges.end();++q) {
		MA[(*q).first+g.nv*(*q).second] = true;
		MA[(*q).second+g.nv*(*q).first] = true;
	}
	std::vector<int> adj(g.nv);
	std::fill(adj.begin(),adj.end(),0);
	for(k=0;k<int(g.nv*g.nv);++k) if(MA[k]) {
		i = k%g.nv; j = k/g.nv;
		if(i<j) { ++adj[i]; ++adj[j]; }
	}
	for(i=1;i<int(g.nv);++i) if(adj[0]!=adj[i])
		return false;
	K = adj[0];
	LAMDA = -1; MU = -1;
	for(i=0;i<int(g.nv)-1;++i) for(j=i+1;j<int(g.nv);++j) {
		int n = 0;
		for(k=0;k<int(g.nv);++k) if(k!=i && k!=j)
			if( MA[i*g.nv+k] && MA[j*g.nv+k] ) ++n;
		if( MA[i*g.nv+j] ) {
			if(LAMDA==-1) LAMDA = n;
			else if(LAMDA!=n )
				return false;
		} else {
			if(MU==-1) MU = n;
			else if(MU!=n )
				return false;
		}
	}
	return true;
}

void make_K(graph & g, size_t n, std::vector<point2> & pos) {
	g.nv = n; g.edges.clear();
	for(size_t i=0;i<n-1;++i)
	for(size_t j=i+1;j<n;++j)
		g.edges.push_back(edge(i,j));
	pos.resize(n);
	for(size_t k=0;k<n;++k) {
		const double a = 2*PI*k/n + PI/2;
		pos[k].x = cos(a);
		pos[k].y = sin(a);
	}
}

void make_line( const graph & g, const std::vector<point2> & ipos, graph & l, std::vector<point2> & opos ) {
	l.nv = g.edges.size();
	l.edges.clear();
	for(size_t i=0;i<g.edges.size()-1;++i)
	for(size_t j=i+1;j<g.edges.size();++j)
		if(g.edges[i].first ==g.edges[j].first  || g.edges[i].first ==g.edges[j].second ||
		   g.edges[i].second==g.edges[j].first  || g.edges[i].second==g.edges[j].second )
				l.edges.push_back( edge(i,j) );
	opos.resize(l.nv);
	for(size_t k=0;k<g.edges.size();++k) {
		opos[k].x = (ipos[g.edges[k].first].x + ipos[g.edges[k].second].x)/2;
		opos[k].y = (ipos[g.edges[k].first].y + ipos[g.edges[k].second].y)/2;
	}
}

void invert( const graph & g, const std::set<size_t> & iset, graph & ig ) {
	size_t i,j;
	std::set<edge> re;
	ig.nv = g.nv; ig.edges.clear();
	for(i=0;i<g.edges.size();++i) {
		bool inf = iset.find(g.edges[i].first )!=iset.end();
		bool ins = iset.find(g.edges[i].second)!=iset.end();
		if(inf ^ ins) re.insert(g.edges[i]);
		else          ig.edges.push_back(g.edges[i]);
	}
	for(i=0;i<g.nv-1;++i)
	for(j=i+1;j<g.nv;++j) {
		bool inf = iset.find(i)!=iset.end();
		bool ins = iset.find(j)!=iset.end();
		if((inf ^ ins) && re.find(edge(i,j))==re.end())
			ig.edges.push_back(edge(i,j));
	}
}

const double SX = 800;
const double SY = 1200;
const double RR = 7;
const double BO = 10;

void save_svg2_start( FILE * fo ) {
	fprintf(fo,
		"<?xml version=\"1.0\" encoding=\"UTF-8\" standalone=\"no\"?>\n"
		"<svg\n"
		"xmlns:svg=\"http://www.w3.org/2000/svg\"\n"
		"xmlns=\"http://www.w3.org/2000/svg\"\n"
		"version=\"1.0\"\n"
		"width=\"%g\"\n" "height=\"%g\"\n"
		"id=\"changgraphs\">\n"
		,SX,SY
	);
}

void save_svg2( FILE * fo, const graph & g, const std::vector<point2> & pos, const std::set<size_t> & rset,
	double ox, double oy ) {
	std::vector<double> px(g.nv);
	std::vector<double> py(g.nv);
	int i;
	const double R = ((SX/2-BO*2)/2);
		
	for(i=0;i<int(g.nv);++i) {
		px[i] = ox+SX/4 + R*pos[i].x;
		py[i] = oy+SY/6 + R*pos[i].y;
	}

	const int ecolor[3][3] = { {0,0,0},{128,0,0},{0,0,64} };
	fprintf(fo,"<g style=\"stroke:#%02X%02X%02X;stroke-width:1.5;stroke-opacity:0.2\">\n"
		,ecolor[2][0],ecolor[2][1],ecolor[2][2]);
	for(int a=0;a<int(g.nv)-1;++a)
	for(int b=a+1;b<int(g.nv);++b)
	{
		bool f1 = rset.find(a)==rset.end();
		bool f2 = rset.find(b)==rset.end();
		if(f1==f2) continue;
		if( g.find_edge( edge(a,b) )==-1 )
			fprintf(fo,
					"<line x1=\"%3.1lf\" y1=\"%3.1lf\" x2=\"%3.1lf\" y2=\"%3.1lf\"/>\n"
					,px[a],py[a]
					,px[b],py[b]
				);
	}
	fprintf(fo,"</g>\n");
	for(int mode=0;mode<2;++mode) {
		fprintf(fo,"<g style=\"stroke:#%02X%02X%02X;stroke-width:1.5;stroke-opacity:0.75\">\n"
			,ecolor[mode][0],ecolor[mode][1],ecolor[mode][2]);
		for(i=0;i<int(g.edges.size());++i)
		{
			bool f1 = rset.find(g.edges[i].first)==rset.end();
			bool f2 = rset.find(g.edges[i].second)==rset.end();
			if( (mode==0 && f1==f2) || (mode==1 && f1!=f2) )
				fprintf(fo,
					"<line x1=\"%3.1lf\" y1=\"%3.1lf\" x2=\"%3.1lf\" y2=\"%3.1lf\"/>\n"
					,px[g.edges[i].first ],py[g.edges[i].first ]
					,px[g.edges[i].second],py[g.edges[i].second]
				);
		}
		fprintf(fo,"</g>\n");
	}

	fprintf(fo,"<g id=\"nodes\" style=\"stroke:#000000;stroke-width:1;fill:#0000FF\">\n");
	for(i=0;i<int(g.nv);++i) if(rset.find(i)==rset.end())
		fprintf(fo,"<circle cx=\"%3.1lf\" cy=\"%3.1lf\" r=\"%g\"/>\n",px[i],py[i],RR);
	fprintf(fo,"</g>\n");
	fprintf(fo,"<g id=\"nodes\" style=\"stroke:#000000;stroke-width:1;fill:#00E000\">\n");
	for(i=0;i<int(g.nv);++i) if(rset.find(i)!=rset.end())
		fprintf(fo,"<circle cx=\"%3.1lf\" cy=\"%3.1lf\" r=\"%g\"/>\n",px[i],py[i],RR);
	fprintf(fo,"</g>\n");

}

void save_svg2_end( FILE * fo ) {
	fprintf(fo,"</svg>\n");
}

int main() {
	size_t i;
	graph K8,T8,chang1,chang2,chang3;
	std::vector<point2> k8_pos,t8_pos;
	std::set<size_t> empty,rset1,rset2,rset3;
	make_K(K8,8,k8_pos);            printf("%u %u\n",K8.nv,K8.edges.size());
	make_line(K8,k8_pos,T8,t8_pos); printf("%u %u\n",T8.nv,T8.edges.size());
	int K,LAMBDA,MU;
	if(!is_strong_regular(T8,K,LAMBDA,MU)) printf("error");
	else printf("t8: %u %d %d %d\n",T8.nv,K,LAMBDA,MU);

	FILE * fo = fopen("c:\\temp\\chang_graphs.svg","w");
	save_svg2_start(fo);

	std::vector<edge> fourK2;
	fourK2.push_back( edge(0,1) ); fourK2.push_back( edge(2,3) );
	fourK2.push_back( edge(4,5) ); fourK2.push_back( edge(6,7) );
	for(i=0;i<fourK2.size();++i) rset1.insert( K8.find_edge( fourK2[i] ) );
	invert(T8,rset1,chang1);
	if(!is_strong_regular(chang1,K,LAMBDA,MU)) printf("error");
	else printf("chang1: %u %d %d %d\n",T8.nv,K,LAMBDA,MU);
	save_svg2(fo,T8,t8_pos,rset1,0,0);
	save_svg2(fo,chang1,t8_pos,empty,SX/2,0);

	std::vector<edge> c8;
	c8.push_back( edge(0,1) ); c8.push_back( edge(1,2) );
	c8.push_back( edge(2,3) ); c8.push_back( edge(3,4) );
	c8.push_back( edge(4,5) ); c8.push_back( edge(5,6) );
	c8.push_back( edge(6,7) ); c8.push_back( edge(0,7) );
	for(i=0;i<c8.size();++i) rset2.insert( K8.find_edge( c8[i] ) );
	invert(T8,rset2,chang2);
	if(!is_strong_regular(chang2,K,LAMBDA,MU)) printf("error");
	else printf("chang2: %u %d %d %d\n",T8.nv,K,LAMBDA,MU);
	save_svg2(fo,T8,t8_pos,rset2,0,SY/3);
	save_svg2(fo,chang2,t8_pos,empty,SX/2,SY/3);

	std::vector<edge> c3uc5;
	c3uc5.push_back( edge(0,3) ); c3uc5.push_back( edge(3,5) );
	c3uc5.push_back( edge(0,5) );
	c3uc5.push_back( edge(1,2) ); c3uc5.push_back( edge(2,4) );
	c3uc5.push_back( edge(4,6) ); c3uc5.push_back( edge(6,7) );
	c3uc5.push_back( edge(1,7) );
	for(i=0;i<c3uc5.size();++i) rset3.insert( K8.find_edge( c3uc5[i] ) );
	invert(T8,rset3,chang3);
	if(!is_strong_regular(chang3,K,LAMBDA,MU)) printf("error\n");
	else printf("chang3: %u %d %d %d\n",T8.nv,K,LAMBDA,MU);
	save_svg2(fo,T8,t8_pos,rset3,0,SY*2/3);
	save_svg2(fo,chang3,t8_pos,empty,SX/2,SY*2/3);

	save_svg2_end(fo);
	fclose(fo);
	return 0;
}

Licensing

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current09:15, 29 September 2010Thumbnail for version as of 09:15, 29 September 2010800 × 1,200 (72 KB)Rocchini{{Information |Description={{en|1=The Chang Graphs. The Chang Graphs are strongly regular graphs with parameters (28,12,6,4). On the right the tree Chang Graphs; these graphs are generated by selecting a proper switching. On the left the originating [Tria

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