SUPPOSE F BE A (3, 6) -fullerene GRAPH WITH N VERTICES, NAMELY A PLANAR 3-REGULAR GRAPH HAVE TRIAGONAL AND HEXAGONAL FACES. IF A BE ADJACENCY MATRIX OF F AND L1,…, LN BE EIGENVALUE OF A WE KNOW THE ENERGY OF A GRAPH IS: E (G) =(FORMULA). IN THIS PAPER WE CONSIDER AN INFINIE CLASS OF fullerene GRAPH WITH 8N VERTICES AND FIND SUITABLE LABELING THAT BE CENTROSYMMETRIC. FINALLY WITH SUITABLE BLOCKING AND BY PROPERTIES OF CENTROSYMMETRIC MATRIX, FIND A UPPER BOUND OF ENERGY OF INFINITE CLASS OF THIS fullerene.

SUPPOSE F BE A (3, 6) -fullerene GRAPH WITH N VERTICES, NAMELY A PLANAR 3-REGULAR GRAPH HAVE TRIAGONAL AND HEXAGONAL FACES. WE KNOW FROM EULER’S FORMULA, NUMBER OF TRIANGLE IN THESE graphs BE FOUR. DIASTANCE BETWEEN TWO X AND Y VERTICES BE LENGTH OF SHORTEST PATH BETWEEN X AND Y, AND HAS BEEN SHOWN D(X,Y). CONSIDERW (F) BE HALF SUM OF DISTANCES BETWEEN VERTICES EXCLUSIVELY AND DEFINE AVERAGE OF DISTANCE OF F BE (FORMULA). GENERALLY THIS INVARIANT PROBLEM IN fullerene graphs BE UNSOLVED. IN THIS PAPER WE CONSIDER AN INFIINITE CLASS OF (3, 6) -fullerene GRAPH WITH 8N VERTICES AND COMPUTE AVERAGE DISTANCE OF THEM. BY THIS VALUE, WE PRESENT SOME BOUND OF INVARIANT OF THIS graphs.

A(R, S) -fullerene GRAPH IS A PLANAR 3 REGULAR GRAPH WITH ONLY CR AND CS FACES, WHERE CN DENOTES A CYCLE OF LENGTH N. IN THIS PAPER THE (3, 6) -fullerene CAYLEY graphs CONSTRUCTED FROM FINITE GROUPS ARE CLASSIFIED. A CHARACTERIZATION OF (4, 6) -fullerene CAYLEY graphs IS ALSO PRESENTED.

LET G BE A GROUP GENERATED BY A FINITE SET S. ASSUME THAT S IS SYMMETRIC, NAMELY S=S-1. THE CAYLEY GRAPH X=C (G, S) IS DEFINED AS FOLLOWS. VERTICES OF X ARE ELEMENTS IN G AND TWO VERTICESG1; G2 2 G ARE ADJACENT IF G1=G2S FOR SOME S 2 S. ALSO LET G1 BE AN (N, K) -GRAPH AND LET G2 BE A (K, K’) -GRAPH WITH V (G2) = [K] = (FORMULA) AND FIX A RANDOMLY NUMBERING JG1 OF G1. THE REPLACEMENT PRODUCT (FORMULA) IS THE GRAPH WHOSE VERTEX SET IS (FORMULA) AND THERE IS AN EDGE BETWEEN VERTICES(V, K) AND (W, L) WHENEVER V=W AND (FORMULA) AND JW G(V) =L. IN THIS NOTE WE STUDY THESE NEW PRODUCT OF graphs AND COMPUTE CAYLEY GRAPH OF SOME NANOSTRUCTURE.

A SET SÍ V (G) IS INDEPENDENT IF NO TWO VERTICES FROM S ARE ADJACENT. THE CARDINALITY OF ANY BIGGEST INDEPENDENT SET INV (G) IS CALLED THE INDEPENDENCE NUMBER OF G AND DENOTED BY (G). IN THIS PAPER, WE COMPUTE INDEPENDENCE NUMBER OF INFINITE CLASSES OF fullerene graphs.

IN THIS PAPER WE DEFINE THREE QUOTIENT graphs OF THE POWER graphs AND STUDY THEIR PROPERTIES AND SOME RELATION BETWEEN THEM.