It is well known that the number of vertices of a graph of diameter two and maximum degree an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si2.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=b2fe643221eb1589062fd8f36940e833" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">dath>an>an>an> is at most an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si3.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=8f1adf511e8fa75e251bcbc46ea4185b" title="Click to view the MathML source">d2+1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">d2+1ath>an>an>an>. The number an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si4.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=9f0a21c18603a414adedf149c799e33e" title="Click to view the MathML source">d2+1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">d2+1ath>an>an>an> is the Moore bound for diameter two. Let an id="mmlsi5" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si5.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=a5e99bbc15a17ad74717477dca0ae1ed" title="Click to view the MathML source">C(d,2)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">C(d,2)ath>an>an>an> denote the largest order of a Cayley graph of degree an id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si6.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=fc4960d3af96d82ded454974bd54b59e" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">dath>an>an>an> and diameter two. Up to now, the only known construction of Cayley graphs of diameter two valid for all degrees an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si7.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=dff911e66ae3d670b0a8bc7fcb9fbbd3" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">dath>an>an>an> is a construction giving an id="mmlsi8" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si8.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=c06b2acb754c243f4ee0afe30ac1a706">ass="imgLazyJSB inlineImage" height="21" width="123" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X14001656-si8.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">C(d,2)>ac>14ac>d2+dath>an>an>an>. However, there is a construction yielding Cayley graphs of diameter two, degree an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si9.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=769397670b2144cff0094423d2083e19" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">dath>an>an>an> and order an id="mmlsi10" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si10.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=3cd020832c496a7f6052f3c641b397e7">ass="imgLazyJSB inlineImage" height="21" width="75" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X14001656-si10.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">d2−O(dac>32ac>)ath>an>an>an> for an infinite set of degrees an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si11.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=f648b208fa57ac71be12f553c123a24e" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">dath>an>an>an> of a special type.
In this article we present, for any integer an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si12.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=7bc5532720ce1b721711fe99896c98a0" title="Click to view the MathML source">d≥4an>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">d≥4ath>an>an>an>, a construction of Cayley graphs of diameter two, degree an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si13.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=31459cbee1e4b3e6fee81b3cd6e75fd7" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">dath>an>an>an> and of order an id="mmlsi14" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si14.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=26e54ccde171017a38c5d806587fdb2f">ass="imgLazyJSB inlineImage" height="21" width="51" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X14001656-si14.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">ac>12ac>d2−tath>an>an>an> for an id="mmlsi15" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si15.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=8324cb52909ae5c88f2ecc60c3acb8fa" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si15.gif" overflow="scroll">dath>an>an>an> even and of order an id="mmlsi16" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si16.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=32b5941c0202abcd8d38715746738478">ass="imgLazyJSB inlineImage" height="21" width="91" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0166218X14001656-si16.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si16.gif" overflow="scroll">ac>12ac>(d2+d)−tath>an>an>an> for an id="mmlsi17" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si17.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=1477811b7458630aa639713fc0fa8c0c" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si17.gif" overflow="scroll">dath>an>an>an> odd, where an id="mmlsi18" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si18.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=5dc4478e048b4d887f465a1d141e4dc4" title="Click to view the MathML source">0≤t≤8an>an class="mathContainer hidden">an class="mathCode">ath altimg="si18.gif" overflow="scroll">0≤t≤8ath>an>an>an> is an integer depending on the congruence class of an id="mmlsi19" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0166218X14001656&_mathId=si19.gif&_user=111111111&_pii=S0166218X14001656&_rdoc=1&_issn=0166218X&md5=6f5ea142a520413123926a63110233f4" title="Click to view the MathML source">dan>an class="mathContainer hidden">an class="mathCode">ath altimg="si19.gif" overflow="scroll">dath>an>an>an> modulo 8.
In addition, we show that, in asymptotic sense, the most of record Cayley graphs of diameter two is obtained by our construction.
