Minimal skew energies of oriented bicyclic graphs without even cycles

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1 214c12 $ Ê Æ Æ 118ò 14Ï Dec., 214 Operations Research Transactions Vol.18 No.4 Ø¹óVã4Uþ f 1 1, Á -G {üëïã. ãgz^>dƒ, k ã, P G σ. k ãg σ UþE s(g σ )½Â G σ ÝAŠýéŠƒÚ. -BnL«º: ê nø¹óvã8ü. ÄBn ã UþlŒüS K. ^k ã UþÈ úªú Û {, n 156Ú155 n 12ž, OBn äk! gúgnuþvã. ' c k ã, Vã, Uþ ã aò O157.5, O êæ aò 5C5, 5C35 Minimal skew energies of oriented bicyclic graphs without even cycles XIAO Mao 1 WANG Wenhuan 1, Abstract Let G be a simple connected graph. By assigning an orientation to each edge of G, we obtained an oriented graph G σ. The skew energy E s(g σ ) of an oriented graph G σ is defined as the sum of the absolute eigenvalues of the skew adjacency matrix for G σ. Let B n be the set of bicyclic graphs without even cycles having n vertices. The ordering of graphs in B n in terms of their minimal skew energies was considered. By employing the integral formula of skew energy and knowledge of real analysis, we deduced the first three graphs with minimal skew energies in B n for n 156 and 155 n 12, respectively. Keywords oriented graphs, bicyclic graphs, skew energy Chinese Library Classification O157.5, O Mathematics Subject Classification 5C5, 5C35 Ú ó -G º: ê n{üã ã, V (G) = {q 1, q 2,, q n } ãgº:8. ãgu þ½â ãgýašýéšƒú [1]. duãuþïä9nõêæ+, 'X êãø!ý! ÜêÆÚ Û, ÏdÚåïÄóŠö2,. éãu þ [), Œ z[1,2]. éãgz^>dƒ, k ãp G σ. G σ Ý,P S(G σ ), ùps(g σ ) = [s ij ]. eq i q j G σ lq i q j ^k l, Ks ij = 1±9s ji = 1. ÄK, ÂvFÏµ * Ä7 8µI[g, ÆÄ7(No ), þ ½ :Æ ïä7(no. S314) 1. þ ŒÆnÆêÆX, þ 2444; Department of Mathematics, College of Sciences, Shanghai University, Shanghai 2444, China ÏÕŠö Corresponding author, whwang@shu.edu.cn

2 86 f, 18ò s ij = s ji =. C, Adiga [3] òãuþvgí2k ãuþ. éuk ãg σ, ½ÂU þ [3] n E s (G σ ) = λ i, (.1) Ù λ 1, λ 2,, λ n S(G σ )AŠ, =φ(g σ, x) = n Š. ùp, φ(g σ, x) = det[xi S(G σ )] = n b i (G σ )x n i G σ Aõ ª, I nü Ý, b, b 1,, b n φ(g σ, x) i= Xê. dug {üã, G σ Ø ¹ Úõ >, S(G σ ) é Ý. u λ i (1 i n n) XJê, éu k i, b 2i (G σ ) ±9b 2i+1 (G σ ) = [3]. Ïd, 2 n 2 i=1 φ(g σ, x) = b 2i (G σ )x n 2i. (.2) ^φ(g σ, x)xê, Hou [4] òuþe s (G σ )Lˆ e È úª: l(.3)œ±wñ, i E s (G σ ) = 2 π i= n 1 [ 2 x 2 log b 2i (G σ )x 2i] dx. (.3) i= n 2, E s (G σ ) b 2i (G σ )î ün4o¼ê. ùpb (G σ ) = 1, b 2 (G σ )uãg>ê. ÚãUþXê'{aq, (.3) 'k ãuþœj ø «š~k^ {. -G σ1 1 ÚGσ2 2 ü nk ã, k n ) b 2i (G σ1 1 ) b 2i(G σ2 2 ( ) i = E s (G σ1 1 2 ) E s(g σ2 2 ). (.4) éu k i n 2, eb 2i(G σ1 1 ) = b 2i(G σ2 2 ), KE s(g σ1 1 ) = E s(g σ2 2 ). (.4) U þxê'{. ^k ãuþxê'{úù{ êãø {, < éäk4šuþk ã?1ïä. éunk üã, Hou [4] äk!gú ŒUþã; Zhu [5] xc n 9 2 äkœuþã. éuk Vã, Shen[6] äk Ú ŒUþã. éuk (n, m)-ã, n m 2(n 2)ž, Gong [7] xäk Uþã. éuù{4šuþã x, Œ z[8-1]. -BnL«º: ê n, Ø¹óVã8Ü. éuø¹óã Â, Œë z[11]. ÄBn ã UþlŒüS K. ^Xê'{Ú Û {, n 156Ú155 n 12ž, OBn cn äk4uþvã? 1üS. 1 OóŠ Ì (Ø, Ú\ 7 Ún, Ñ ã½â. éug σ óc l, ec l kóê^> ƒó(=óž ^ž½_ž ), K C l ó½ ; ÄK C l Û½. ãg 5fãL ãg p

3 4Ï Ø¹óVã4Uþ 87 Øƒ>Ú. elø ¹Û, K L ó 5fã. -EL 2i (G) ãg º: ê 2ió 5fã8Ü. éu 5fãL EL 2i (G), -P e (L)ÚP o (L) OL«L ó ½ ÚÛ½ ê8. HouÚLei [12] e Ún1.1. Ún1.1 [12] -G σ L«k ã, K b 2i (G σ ) = L EL 2i(G σ ) ( 2) Pe(L) 2 Po(L), (1.1) Ù P e (L) ó½ ê8, P o (L) Û½ ê8. ŠâÚn1.1, ek ãg σ Ø¹ó, Kb 2i (G σ ) = m(g, i). Ún1.2 [5] -φ(g σ1 1, x)úφ(gσ2 2, x) O nk ãgσ1 1 ÚGσ2 2 Aõ ª, K E s (G σ1 1 ) E s(g σ2 2 ) = 2 π log φ(gσ1 1, x) (1.2) φ(g σ2, x)dx. ãg kšl«ãg k^pøƒ>. -m(g, k)l«ãgkšê 8. k =, ½Âm(G, ) = 1. k <, m(g, k) =. Ún1.3 [1] -e = uv ãg ^>, k m(g, k) = m(g e, k) + m(g u v, k 1). (1.3) éuü ãg 1 ÚG 2, ½Âe šê[s'x. éu kk, em(g 1, k) m(g 2, k), KPG 1 G 2. XJG 1 G 2, 3, êk, m(g 1, k ) < m(g 2, k ), KPG 1 G 2. eg 1 G 2 G 2 G 1 ÑØ á, K G 1 ÚG 2 ØŒ'. -P n L«n º:. PP n n º: v 1, v 2,, v n. -X n L«(ãK 1,n 1, Y n L «3P 4 º:v 2 þñn 4^]!> ã, Z n L«3P 4 º:v 2 Úv 3 þ O Ñn 5^Ú1^]!> ã. Ún1.4 [13] -T n º:ä, n 6, KX n Y n Z n T, ùpt X n, Y n, Z n. -B n,d L«3P d+1 º:v 2 þñn d 1^]!> ã, B n L«3P 6 º :v 3 þñn 6^]!> ã, C n L«3P 6 º:v 2 Úv 5 þ OÑn 7^Ú1^ ]!> ã. Ún1.5 [14] -T äkn º:»Øud 1ä, KB n,d T, Ò á =T = B n,d. Ún1.6 [15] T äkn º:» 5ä, Ù n 8, KC n T, ùpt B n,5, B n, C n. Ún1.7 [16] éu êx > 1, k X 1+X log(1 + X) X. Ún1.8 [17] eg Ø¹óëÏã, KG? ü Û >ƒpø. 2 2 Ì (J -G B n. PG ü Û C a ÚC b, Ù a, b 3, b a. C a ÚC b º: ^ž OP u 1, u 2,, u a Úw 1, w 2,, w b. dún1.8, C a ÚC b mvkúº:½ k

4 88 f, 18ò úº:. 3Ún2.2ÚÚn2.3, OÄùü«œ¹. e kú\ún2.1. éuëïãg, -d(g)l«ãg». Ún2.2ÚÚn2.3, Ún2.1 -G B n, n 8. 3G, -e 1 Úe 2 O C a ÚC b ^>, K (i) d(g e 1 e 2 ) 6, b 4 (G σ ) 4n 18, b 6 (G σ ) 3n 17; (ii) d(g e 1 e 2 ) = 5 G e 1 e 2 B n,5, B n, b 4 (G σ ) 4n 19, b 6 (G σ ) 2n 12. y² -G Bn, n 8. 3G, ee 1 Úe 2 O C a ÚC b ^>, KdÚn1.8, G e 1 e 2 º: ê nä. qšâún1.1, kb 2i (G σ ) = m(g, i). (i) d(g e 1 e 2 ) 6, ŠâÚn1.3ÚÚn1.5, k b 4 (G σ ) = m(g, 2) m(g e 1 e 2, 2) m(b n,6, 2) = 4n 18, b 6 (G σ ) = m(g, 3) m(g e 1 e 2, 3) m(b n,6, 3) = 3n 17. (ii) d(g e 1 e 2 ) = 5 G e 1 e 2 B n,5, B n, kn 8. ŠâÚn1.3ÚÚn1.6, k b 4 (G σ ) = m(g, 2) m(g e 1 e 2, 2) m(c n, 2) = 4n 19, b 6 (G σ ) = m(g, 3) m(g e 1 e 2, 3) m(c n, 3) = 2n 12. Ún2.2 -G B n, n 8. 3G, ec a C b vkúº:, Kb 4 (G σ ) 4n 19, b 6 (G σ ) 2n 12. y² -G B n, n 8. 3G, ec a C b vkúº:, KC a C b ƒmk ^ P. Ø bp ü à: O u 1 Úw 1, P Ý l(p ). w,, l(p ) 1. l(p ) 2 a, b 3½öl(P ) = 1 a 3Úb 5, G u 1 u a w 1 w b u 1 u a w 1 w b ) 6. dún2.1(i), Ún2.2 á. -l(p ) = 1 a = b = 3. e{u 2, u 3, w 2, w 3 } k :(Ø Kd(G u 1 u 3 w 1 w 3 ) 6. dún2.1(i), Ún2.2 á. ä d(g w 3 )!Ñ ˆä, e b{u 2, u 3, w 2, w 3 } k:ñø!ñä. eu 1 ½w 1 (Ø u 1 )þñ ^ Ý 2, Kd(G u 1 u 3 w 1 w 3 ) 5 G u 1 u 3 w 1 w 3 B n,5, B n. dún1.3, Ú n2.2 á. e b½u 1 Úw 1 O!Ñx^Úy^]!>, Ù x, y x + y = n 6. dún1.3, ²L OŽ, n 8, kb 4 (G σ ) = m(g, 2) = (x + y) + (3 + x)(3 + y) + 2 = 4n 13 + xy > 4n 19, b 6 (G σ ) = m(g, 3) = x(3 + y) + y(3 + x) + 1 = 3n xy > 2n 12. Ïd, Ú n2.2 á. -S 5,3,3 L«ò C 3 º:u 1, C 3 º:w 1?1ÅÜ ã. - A n,3,3 L«3S 5,3,3 º:u 1 þñn 5^]!> ã, B n,3,3 L«3S 5,3,3 º:u 1 þ Ñn 7^]!>Ú ^ 2 ã, C n,3,3 L«3S 5,3,3 º:u 1 Úu 2 þ O Ñn 6^Ú1^]!> ã, D n,3,3 L«3S 5,3,3 º:u 2 þñn 5^]!> ã. A n,3,3, B n,3,3, C n,3,3 ÚD n,3,3 OXã1 «.

5 4Ï Ø¹óVã4Uþ 89 ã1 A n,3,3, B n,3,3, C n,3,3, D n,3,3 Ún2.3 -G Bn, n 8. 3G, ec a C b k úº: G A n,3,3, B n,3,3, C n,3,3, D n,3,3, Kb 4 (G σ ) 4n 19, b 6 (G σ ) 2n 12. y² -G Bn, n 8. 3G, ec a C b k úº:, KØ C a º :u 1 C b º:w 1 úº:, Ù b a. w,, G u 1 u a w 1 w b n º:ä. b 5, a 3, kd(g u 1 u a w 1 w b ) 6. dún2.1(i), Ún2.3 á. e -b = a = 3. e{u 2, u 3, w 2, w 3 } 3 º:(Ø u 2 )Ñ ^Ý 2, KG u 1 u 2 w 1 w 3 n º:ä d(g u 1 u 2 w 1 w 3 ) 6. dún2.1(i), Ún2.3 á. e, e{u 2, u 3, w 2, w 3 } º:Ñä, Kb½ùˆä (ã. Šâ{u 2, u 3, w 2, w 3 } º: ÄÑ(ã n«œ¹?1?ø. œ¹1 {u 2, u 3, w 2, w 3 } kº:ñøñ(ã. -e 1 = u 2 u 3, e 2 = w 2 w 3, KG e 1 e 2 n º:ä. eg A n,3,3, B n,3,3, KG e 1 e 2 X n, Y n. ŠâÚn1.4, km(g e 1 e 2, 2) m(z n, 2) = 2n 8. ŠâÚn1.1Ú Ún1.3, k b 4 (G σ ) = m(g, 2) = m(g e 1, 2) + m(g u 2 u 3, 1) = [m(g e 1 e 2, 2) + (n 3)] + (n 2) 4n 13 > 4n 19. (2.1) Ï u 2, u 3, w 2 Úw 3 ÑvkÑ(ã, ±G e 1 w 2 w 3 n 2 º:ä. eg A n,3,3, B n,3,3, KG e 1 w 2 w 3 X n 2, Y n 2. ŠâÚn1.4, km(g e 1 w 2 w 3, 2) m(z n 2, 2) = 2n 12. ŠâÚn1.1ÚÚn1.3, k b 6 (G σ ) = m(g, 3) = m(g e 1, 3) + m(g u 2 u 3, 2) = [m(g e 1 e 2, 3) + m(g e 1 w 2 w 3, 2)] + m(g u 2 u 3, 2) m(g e 1 w 2 w 3, 2) 2n 12. (2.2) d(2.1)ú(2.2), Ún2.3 á. œ¹2 {u 2, u 3, w 2, w 3 } k º:(Ø u 2 )Ñ(ã. eu 2 þ Ñ ^]!> G C n,3,3, Ku 1 þñ ^Ý 2. ÏdG u 1 u 2 w 1 w 2 n º:ä, d(g u 1 u 2 w 1 w 2 ) 5 G u 1 u 2 w 1 w 2 B n,5, B n. dú n2.1, Ún2.3 á. eu 2 þ Ñü^]!> G D n,3,3, Ku 1 þñ ä. ÏdG u 1 u 2 w 1 w 2 n º:ä, d(g u 1 u 2 w 1 w 2 ) 5 G u 1 u 2 w 1 w 2 B n,5, B n. dú n2.1, Ún2.3 á.

6 9 f, 18ò œ¹3 {u 2, u 3, w 2, w 3 } kü º:Ñ(ã. Äk, Ø b½u 2 ]!>. ew 2 þñ]!>, KG u 1 u 2 w 1 w 2 n º: ä d(g u 1 u 2 w 1 w 2 ) 6. dún2.1(i), Ún2.3 á. Ón, ew 3 þñ]!>, Ú n2.3 á. ew 2 Úw 3 ÑØÑ]!>, du{u 2, u 3, w 2, w 3 } kü º:Ñ(ã, Ïdu 3 Ñ]!>. XJu 1 Ñ ä, KG u 1 u 2 w 1 w 2 n º:ä, d(g u 1 u 2 w 1 w 2 ) 5 G u 1 u 2 w 1 w 2 B n,5, B n. dún2.1, Ún2.3 á. XJu 1 þvk Ñä, Kn 8, u 2 Úu 3 k º:Ñü^]!>. Ø bù º: u 2, KG u 1 u 2 w 1 w 3 n º:ä, d(g u 1 u 2 w 1 w 3 ) 5 G u 1 u 2 w 1 w 3 B n,5, B n. dún2.1, Ún2.3 á. Ø bbn 3 ãg n,, v b 2 (G σ n,) = n + 1, b 4 (G σ n,) = 4n 19, b 6 (G σ n,) = 2n 12, b 2i (G σ n,) =, Ù 8 2i n. dún1.8!ún2.2úún2.3, ŒÚn2.4. Ún2.4 -G Bn, Ù n 8, ke s (G σ n,) E s (G σ ), Ù G A n,3,3, B n,3,3, C n,3,3, D n,3,3. y² -G Bn, Ù n 8. dugø¹ó, dún1.8, C a ÚC b mvkúº: ½ k úº:. eg A n,3,3, B n,3,3, C n,3,3, D n,3,3, dún2.2úún2.3, k b 2 (G σ ) = b 2 (G σ n,) = n + 1, b 6 (G σ ) 2n 12 = b 6 (G σ n,), b 4 (G σ ) 4n 19 = b 4 (G σ n,), b 2i (G σ ) = b 2i (G σ n,), Ù 8 2i n. d(.4), Ún2.4y. 'A σ n,3,3, Bn,3,3, σ Cn,3,3, σ Dn,3,3ÚG σ σ n,ƒmuþœ, e Ú\Ún Ún y², A^k ãuþxê'{ú Û {, éü k ãuþ?1'. Ún2.5 n 8, (i) E s (A σ n,3,3) < E s (Cn,3,3) σ < E s (Bn,3,3); σ (ii) E s (A σ n,3,3) < E s (Dn,3,3). σ y² n 8, ÏL OŽ, k b 2 (A σ n,3,3) = b 2 (C σ n,3,3) = b 2 (B σ n,3,3) = n + 1, b 4 (A σ n,3,3) = 2n 5 < 3n 9 = b 4 (C σ n,3,3) < 3n 8 = b 4 (B σ n,3,3), b 6 (A σ n,3,3) = n 5 < 2n 11 = b 6 (C σ n,3,3) < 3n 15 = b 6 (B σ n,3,3), b 8 (A σ n,3,3) = b 8 (C σ n,3,3) = < b 8 (B σ n,3,3) = n 7, b 2i (A σ n,3,3) = b 2i (C σ n,3,3) = b 2i (B σ n,3,3) =, Ù 1 2i n. Šâ(.4), Ún2.5(i) á. OŽ, kb 2 (Dn,3,3) σ = n+1, b 4 (Dn,3,3) σ = 4n 15, b 6 (Dn,3,3) σ = n 5, b 2i (Dn,3,3) σ =, ùp8 2i n. aqún2.5(i)y², Ún2.5(ii) á. Ún2.6 n 156, E s (Cn,3,3) σ < E s (Dn,3,3). σ 155 n 8, E s (Cn,3,3) σ > E s (Dn,3,3). σ y² Šâ(.2)ÚÚn1.1, k φ(c σ n,3,3, x) = x n 6 [x 6 + (n + 1)x 4 + (3n 9)x 2 + (2n 11)] x n 6 C(n, x), (2.3) φ(d σ n,3,3, x) = x n 6 [x 6 + (n + 1)x 4 + (4n 15)x 2 + (n 5)] x n 6 D(n, x). (2.4)

7 4Ï Ø¹óVã4Uþ 91 Šâ(2.3), (2.4)ÚÚn1.2, k E s (Cn,3,3) σ E s (Dn,3,3) σ = 2 f(x, n)dx, (2.5) π ( ) ùpf(x, n) = log C(n,x) A+B D(n,x) = log A+C = log 1 + B C A+C, Ù A = x 6 + (n + 1)x 4, B = (3n 9)x 2 + (2n 11), C = (4n 15)x 2 + (n 5), =f(x, n)œ±l«( (n 6) (n 6)x 2 ) f(x, n) = log 1 + x 6 + (n + 1)x 4 + (4n 15)x 2. (2.6) + (n 5) w,/, n 8 x, A, B, C >. -X = B C. yx > 1. ŠâÚn1.7, A + C x R, x n 8ž, k ±9 f(x, n) f(x, n) (n 6) (n 6)x 2 x 6 + (n + 1)x 4 + (4n 15)x 2 + (n 5) (n 6) (n 6)x 2 x 6 + (n + 1)x 4 + (3n 9)x 2 + (2n 11). x 1ž, f(x, n) ; < x < 1ž, f(x, n) >. f(x, n) Âñ, -ϕ(x) = log x4 + 3x x 4 + 4x ùp, 5, x ž, ¼êS lim f(x, n) = log x4 + 3x n + x 4 + 4x (2.7) ¼êSf(x, n)4. x ž, k ϕ(x) f(x, n) = log x4 + 3x x 4 + 4x log x6 + (n + 1)x 4 + (3n 9)x 2 + (2n 11) x 6 + (n + 1)x 4 + (4n 15)x 2 + (n 5) ( = log 1 + P (x, n) Q(x, n) ). (2.8) Q(x, n) P (x, n) = x 1 + (n + 4)x 8 + (7n 1)x 6 + (15n 48)x 4 + (11n 45)x 2 + (2n 1), Q(x, n) = x 1 + (n + 5)x 8 + (7n 4)x 6 + (15n 46)x 4 + (11n 53)x 2 + (2n 11). Ïdk P (x, n) Q(x, n) = 1 + 8x 2 2x 4 6x 6 x 8 = (1 x 2 )(x 6 + 7x 4 + 9x 2 + 1). k x R, n 8ž, P (x, n) Q(x, n) Q(x, n) P (x, n) Q(x, n) ê > 1. ŠâÚn1.7, Q(x, n) ϕ(x) f(x, n) (1 x2 )(x 6 + 7x 4 + 9x 2 + 1) Q(x, n)

8 92 f, 18ò ±9 ϕ(x) f(x, n) (1 x2 )(x 6 + 7x 4 + 9x 2 + 1). P (x, n) Ïd, x 1ž, ϕ(x) f(x, n) ; < x < 1ž, ϕ(x) > f(x, n) >. bn ëycþ. 3f(x, n) én, k f(x, n) n = (1 x2 )(x 6 + 7x 4 + 9x 2 + 1). (2.9) (A + B)(A + C) l(2.9)œ, x 1ž, f(x, n) nün4~¼ê; < x < 1ž, f(x, n) nü N4O¼ê. u, x 1ž, kϕ(x) f(x, n + 1) < f(x, n); < x < 1ž, kϕ(x) > f(x, n + 1) > f(x, n). Ïd, n 164ž, k f(x, n)dx = < 1 1 f(x, n)dx + ϕ(x)dx f(x, n)dx f(x, 164)dx = =. 218 <. (2.1) Šâ(2.5)Ú(2.1)Œ, n 164ž, ke s (Cn,3,3) σ < E s (Dn,3,3). σ ÏL OŽ, 163 n 156, ke s (Cn,3,3) σ < E s (Dn,3,3); σ 155 n 8, ke s (Cn,3,3) σ > E s (Dn,3,3). σ Ú n2.6 á. Ún2.7 n 8, E s (Dn,3,3) σ < E s (Bn,3,3). σ y² Šâ(.2)ÚÚn1.1, k φ(b σ n,3,3, x) = x n 8 [x 8 + (n + 1)x 6 + (3n 8)x 4 + (3n 15)x 2 + (n 7)]. (2.11) Šâ(2.4), (2.11)ÚÚn1.2, k E s (D σ n,3,3) E s (B σ n,3,3) = 2 π ùpf(x, n) = log B(x, n), Ù B(x, n) = ¼êSf(x, n) Âñ, f(x, n)dx, (2.12) x 8 +(n+1)x 6 +(4n 15)x 4 +(n 5)x 2 x 8 +(n+1)x 6 +(3n 8)x 4 +(3n 15)x 2 +(n 7). x ž, lim f(x, n) = log x 6 + 4x 4 + x 2 n + x 6 + 3x 4 + 3x ϕ(x) = log A(x) ¼êSf(x, n)4, Ù A(x) = x6 +4x 4 +x 2 x 6 +3x 4 +3x w,, k ϕ(x) f(x, n) = log A(x) B(x, n). (2.13) ²OŽ, k A(x) B(x, n) = x 2 (x 6 + 4x 4 9x 2 2) (x 2 + 1) 3 [x 4 + (n 1)x 2 + (n 7)]. (2.14)

9 4Ï Ø¹óVã4Uþ 93 -q(x) = x 6 + 4x 4 9x 2 2Úx = ÏL OŽ, < x < x, kq(x) < ; x x, kq(x). d(2.14), < x < x, k < A(x) < B(x, n); x x, ka(x) B(x, n) >, Ù n 8. Ïd, d(2.13)œ, < x < x, ϕ(x) < f(x, n); x x, ϕ(x) f(x, n). OŽ, Œ B(x, n + 1) B(x, n) = x 2 q(x) (x 2 + 1) 2 [x 4 + nx 2 + (n 6)][x 4 + (n 1)x 2 + (n 7)]. -n 8. < x < x, duq(x) <, < B(x, n + 1) < B(x, n); x x, duq(x), B(x, n + 1) B(x, n) >.? Ú, d(2.13)œ, < x < x, f(x, n + 1) < f(x, n); x x, f(x, n+1) f(x, n). Ïd, < x < x, ϕ(x) < f(x, n+1) < f(x, n); x x, ϕ(x) f(x, n + 1) f(x, n). n 8, k f(x, n)dx = = < x x x + x f(x, n)dx + f(x, 8)dx + ( log 1 + x log x x f(x, n)dx ϕ(x)dx x 4 6x 2 1 x 8 + 9x x 4 + 9x ( 1 + x 4 2x 2 1 x 6 + 3x 4 + 3x x 4 6x 2 1 x 8 + 9x x 4 + 9x dx ) dx ) dx x 4 2x x x 6 + 3x 4 + 3x dx = = <, (2.15) (2.15)ê1 uò dún1.7. Ïd, l(2.12)ú(2.15)œ, n 8, E s (Dn,3,3) σ < E s (Bn,3,3). σ Ún2.7 á. Ún2.8 n 12, E s (Cn,3,3) σ < E s (G σ n,). 11 n 8, E s (Cn,3,3) σ > E s (G σ n,). y² Šâ(.2)ÚÚn1.1, k φ(g σ n,, x) = x n 6 [x 6 + (n + 1)x 4 + (4n 19)x 2 + (2n 12)] Šâ(2.16), (2.13)ÚÚn1.2, k -H 1 (x, n) = G(n,x) C(n,x). bn x n 6 G(n, x). (2.16) E s (G σ n,) E s (C σ n,3,3) = 2 π H 1 (x, n) n ëycþ, K log = x8 + 11x x x C 2 (n, x) G(n, x) dx. (2.17) C(n, x) >. (2.18)

10 94 f, 18ò Šâ(2.18)Œ, H 1 (x, n) nün4o¼ê. nün4o¼ê. n 12, ÏLOŽ, k Ïd, Šâ(2.17), E s (G σ n,) E s (C σ n,3,3) E s (G σ n,) E s (C σ n,3,3) E s (G σ 12,) E s (C σ 12,3,3) = 2 G(12, x) log π C(12, x) dx = >. (2.19) d(2.19), n 12, Ún2.8y. 11 n 8, ÏL OŽ, ke s (Cn,3,3) σ > E s (G σ n,). Ún2.9 n 9, E s (Dn,3,3) σ < E s (G σ n,). n = 8, E s (Dn,3,3) σ > E s (G σ n,). y² Šâ(2.16), (2.14)ÚÚn1.2, k E s (G σ n,) E s (D σ n,3,3) = 2 π log G(n, x) dx. (2.2) D(n, x) -H 2 (x, n) = G(n,x) D(n,x). bn ëycþ, K H 2 (x, n) n = 5x6 + 24x x D 2 (n, x) >. (2.21) Ïd, Šâ(2.2)Ú(2.21), E s (G σ n,) E s (D σ n,3,3) nün4o¼ê. n 9, ÏLOŽ, k E s (G σ n,) E s (D σ n,3,3) E s (G σ 9,) E s (D σ 9,3,3) = 2 G(9, x) log π D(9, x) dx = >. (2.22) dª(2.22), n 9, Ún2.9y. n = 8, ÏL OŽ, ke s (Dn,3,3) σ > E s (G σ n,). dún , Bn äk!gúgnuþk ã. ½n2.1 -G Bn. (i) n 156, Ù G A n,3,3, C n,3,3, D n,3,3. (ii) 155 n 12, E s (A σ n,3,3) < E s (C σ n,3,3) < E s (D σ n,3,3) < E s (G σ ), E s (A σ n,3,3) < E s (D σ n,3,3) < E s (C σ n,3,3) < E s (G σ ), Ù G A n,3,3, D n,3,3, C n,3,3. y² (i) n 156. dún2.4!ún2.5(i)!ún2.6úún2.9, k E s (A σ n,3,3) < E s (C σ n,3,3) < E s (D σ n,3,3) < E s (G σ n,) E s (G σ ),

11 4Ï Ø¹óVã4Uþ 95 Ù G A n,3,3, B n,3,3, C n,3,3, D n,3,3. qdún2.7, k E s (D σ n,3,3) < E s (B σ n,3,3). ½n2.1(i)y. (ii) 155 n 12. dún2.4!ún2.5(ii)!ún2.6úún2.8, k E s (A σ n,3,3) < E s (D σ n,3,3) < E s (C σ n,3,3) < E s (G σ n,) E s (G σ ), Ù G A n,3,3, B n,3,3, C n,3,3, D n,3,3. qdún2.5(i), k ½n2.1(ii)y. ë z E s (C σ n,3,3) < E s (B σ n,3,3). [1] Gutman I, Polansky O E. Mathematical Concepts in Organic Chemistry [M]. Berlin: Springer- Verlag, [2] Li X L, Shi Y T, Gutman I. Graph Energy [M]. New York: Springer, 212. [3] Adiga C, Balakrishnan R, So W. The skew energy of a digraph [J]. Linear Algebra and Its Applications, 21, 432(7): [4] Hou Y P, Sun X N, Zhang C Y. Oriented unicyclic graphs with extremal skew energy [EB/OL]. [ ]. [5] Zhu J M. Oriented unicyclic graphs with the first largest skew energies [J]. Linear Algebra and Its Applications, 212, 437(1): [6] Shen X L, Hou Y P, Zhang C Y. Bicyclic digraphs with extremal skew energy [J]. Electronic Journal of Linear Algebra, 212, 23: [7] Gong S C, Li X L, Xu G H. On oriented graphs with minimal skew energy [J]. Electronic Journal of Linear Algebra, 214, 47: [8] Gong S C, Xu G H. 3-regular digraphs with optimum skew energy [J]. Linear Algebra and Its Applications, 212, 436: [9] Chen X L, Li X L, Lian H S. 4-Regular oriented graphs with optimum skew energy [J]. Linear Algebra and Its Applications, 213, 439(1): [1] Tian G X. On the skew energy of orientations of hypercubes [J]. Linear Algebra and Its Applications, 211, 435(9): [11] Anuradha A, Balakrishnan R, So W. Skew spectra of graphs without even cycles [J]. Linear Algebra and Its Applications, 214, 444(1): [12] Hou Y P, Lei T G. Characteristic polynomials of skew-adjacency matrices of oriented graphs [J]. The Electronic Journal of Combinatorics, 211, 18(1): [13] Gutman I. Acyclic systems with extremal Huckel π-electron energy [J]. Theoretica Chimica Acta, 1977, 45: [14] Yan W G, Ye L Z. On the minimal energy of trees with a given diameter [J]. Applied Mathematics Letters, 25, 18(9): [15] Zhou B, Li F. On minimal energies of trees of a prescribed diameter [J]. Journal of Mathematical Chemistry, 26, 39(3/4): [16] Huo B F, Li X L, Shi Y T, et al. Determining the conjugated trees with the third through the sixth minimal energies [J]. Match-Communications in Mathematical and in Computer Chemistry, 21, 65: [17] Wang W H, Xu W W. Graphs with the maximal Estrada indices [J]. Linear Algebra and Its Applications, 214, 446:

The skew-rank of oriented graphs

The skew-rank of oriented graphs Xueliang Li a, Guihai Yu a,b, a Center for Combinatorics and LPMC-TJKLC arxiv:14047230v1 [mathco] 29 Apr 2014 Nankai University, Tianjin 300071, China b Department of Mathematics