JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE)

*-zip rings

Yao WANG,Xin LI,Yanli REN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 1-7. doi:10.6040/j.issn.1671-9352.0.2022.489

Abstract ( 234 )

HTML ( 6 )

PDF (651KB) ( 298 )

Save

References | Related Articles | Metrics

The concept of *-zip ring is introduced. Some examples of this class rings are given and their extension properties are investigate, prove that (1) Let R be a *-ring, n ∈ N and n≥2, then R is *-zip if and only if V_n(R) is * -zip; (2) Let R be a *-skew Armendariz ring, then R is *-zip if and only if *-skew polynomial ring R[x; *] is *-zip.

Gorenstein FP-injective dimensions over formal triangular matrix rings

Cuiping ZHANG,Jiaojiao DONG,Yinyin YANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 8-13. doi:10.6040/j.issn.1671-9352.0.2022.551

Abstract ( 224 )

HTML ( 3 )

PDF (633KB) ( 257 )

Save

References | Related Articles | Metrics

Let $T=\left(\begin{array}{ll}A & 0 \\U & B\end{array}\right)$ be a formal triangular matrix ring, where A and B are rings and U is a (B, A)-bimodule. Let $M=\left(\begin{array}{l}M_1 \\M_2\end{array}\right)_{\varphi^M}$ be a left T-module. The results are proved that if T is a left GFPI-closed and left coherent ring, U_A is flat, _BU is finitely presented and pd(_BU)<∞, then:(1) max{G-FP-id(M₁), G-FP-id(M₂)}≤G-FP-id(M); (2) G-FP-id(M)≤max{G-FP-id(M₁), G-FP-id(M₂)+1};(3) max{lG-FP-id(A), lG-FP-id(B)}≤lG-FP-id(T)≤max{lG-FP-id(A), lG-FP-id(B)+1}.

The module algebra structures of dihedral groups on the full matrix algebra of 2×2

Dong SU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 14-21, 31. doi:10.6040/j.issn.1671-9352.0.2022.632

Abstract ( 221 )

HTML ( 2 )

PDF (635KB) ( 188 )

Save

References | Related Articles | Metrics

By using the theories of weak similarity of matrices and weak similarity of matrix pairs, the module algebra structures of cyclic groups of C_m are researched over the full matrix algebra of 2×2 on the complex field $\mathscr{C}$, and all module algebra structures of dihedral groups D_2m are characterized over the full matrix algebra of 2×2 up to isomorphism.

On w-IFP-flat modules and w-IFP-injective modules

Liu ZHOU,Lei QIAO,Dan ZHAO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 22-31. doi:10.6040/j.issn.1671-9352.0.2022.470

Abstract ( 206 )

HTML ( 6 )

PDF (690KB) ( 365 )

Save

References | Related Articles | Metrics

The w-module theoretic analogues of IFP-flat modules and IFP-injective modules are studied in terms of w-module theory over commutative rings. The concepts of w-IFP-flat modules and w-IFP-injective modules over commutative rings are introduced and several characterizations and properties of these modules are discussed.

Cohomology and deformation of differential Lie-Yamaguti superalgebras

Wen TENG,Fengshan LONG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 32-37,46. doi:10.6040/j.issn.1671-9352.0.2022.665

Abstract ( 186 )

HTML ( 3 )

PDF (689KB) ( 350 )

Save

References | Related Articles | Metrics

Firstly, the representation and cohomology of differential Lie-Yamaguti superalgebras are introduced. Then, the linear deformation of differential Lie-Yamaguti superalgebras is considered according to cohomology.

E-total coloring of wheels and fans vertex-distinguished by multiple sets

Jing CAO,Xiang'en CHEN

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 38-46. doi:10.6040/j.issn.1671-9352.0.2022.591

Abstract ( 202 )

HTML ( 2 )

PDF (1497KB) ( 337 )

Save

Figures and Tables | References | Related Articles | Metrics

By using the method of contradiction and the method of constructing concrete coloring, we discuss E-total coloring of wheels and fans which are vertex-distinguished by multiple sets. The methods of the corresponding colorings of wheels and fans are obtained, and the E-total coloring algorithm of wheels which are vertex-distinguished by multiple sets are constructed.

2-distance sum distinguishing coloring of trees

Huan LIU,Huiying QIANG,Hongshen WANG,Yu BAI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 47-52, 58. doi:10.6040/j.issn.1671-9352.0.2022.347

Abstract ( 188 )

HTML ( 2 )

PDF (764KB) ( 375 )

Save

Figures and Tables | References | Related Articles | Metrics

Based on the structural characteristics of the trees, the 2-distance sum distinguishing edge(total) coloring of trees are studied by using the mathematical induction, combination analytic method and Combinatorial Nullstellensatz, and the 2-distance sum distinguishing edge(total) chromatic numbers are obtained.

Strong edge-coloring of graphs without K_{1, 3}⁺

Jiaxin YUAN,Mingfang HUANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 53-58. doi:10.6040/j.issn.1671-9352.0.2022.557

Abstract ( 223 )

HTML ( 2 )

PDF (640KB) ( 146 )

Save

Figures and Tables | References | Related Articles | Metrics

The strong edge-coloring of a graph G is to assign colors to all edges, so that the derived subgraphs of each color class are a matching. The minimum number of colors required in the strong edge-coloring of a graph G is called the strong chromatic index of the graph G, the degree of edge e=uv is recorded as d(e)=d(u)+d(v), the edge degree of G is recorded as d(G)=min{d(e)|e ∈ E(G)}. This paper proves that the strong chromatic index of the graph without K_{1, 3}⁺ with the maximum degree Δ and edge degree of the graph greater than the number of vertices is at most Δ²-Δ+1.

On r-hued coloring of W_n□P_m

Yaxin SHI,Fengxia LIU,Hua CAI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 59-64. doi:10.6040/j.issn.1671-9352.0.2022.654

Abstract ( 158 )

HTML ( 2 )

PDF (617KB) ( 311 )

Save

References | Related Articles | Metrics

A(k, r)-coloring of a graph G is a proper k-coloring of graph G such that the neighbors of any vertex receive at least min{r, d(v)} different colors. The smallest positive integral k such that graph G has a(k, r)-coloring is defined as the r-hued chromatic number and denoted by χ_r(G). The Cartesian product of two graphs G and H, denoted by G□H, has vertex set V(G)×V(H), where(u₁, v₁) and(u₂, v₂) are adjacent if and only if either u₁=u₂ and v₁v₂∈E(G), or v₁=v₂ and u₁u₂∈E(G). In this paper, the r-hued chromatic number of W_n□P_m is determined.

Construction of singularly cospectral graphs

Chaofan LIANG,Fenjin LIU,Yuchao LI,Shunyi LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 65-70. doi:10.6040/j.issn.1671-9352.0.2022.589

Abstract ( 180 )

HTML ( 2 )

PDF (3414KB) ( 378 )

Save

Figures and Tables | References | Related Articles | Metrics

Two graphs are called singularly cospectral if they have the same nonzero singular value with the same multiplicity. Singularly cospectral is weaker than cospectral but stronger than equienergetic. A new construction of singularly cospectral graph is derived using the operation of t-join(shadow) corona of a graph and the technique of block matrix which has great significance for studying the structural and spectral properties of equienergetic graphs.

Equitable total coloring of generalized Petersen graphs P(n, k)

Yanan SU,Chunling TONG,Yong LI,Senyuan SU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 71-79. doi:10.6040/j.issn.1671-9352.0.2022.413

Abstract ( 212 )

HTML ( 1 )

PDF (3168KB) ( 278 )

Save

Figures and Tables | References | Related Articles | Metrics

Generalized Petersen graphs P(n, k) are the most widely studied in coloring problems. However, the total chromatic number of P(n, k) for k(mod 4)=0 needs to be further studied. By combining computer searching and mathematics techniques, the equitable total chromatic number of P(n, k) for k(mod 16)=4, 8, 12 and k(mod 16)=0∧n(mod 2k)=0, 1, 2, 4 are obtained.

Study on semi-paired k-disjoint path cover of unit interval graphs

Li ZHU,Peng LI,Aifa WANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 80-90. doi:10.6040/j.issn.1671-9352.0.2022.404

Abstract ( 192 )

HTML ( 1 )

PDF (3942KB) ( 349 )

Save

Figures and Tables | References | Related Articles | Metrics

The fault tolerance problem of semi-paired, many to many and k-disjoint path cover (k-DPC) on unit interval graphs is studied. Using the structural characteristics of the path cover and the structural properties of the vertex order of the unit interval graphs, the unit interval graphs with semi-paired 1-DPC and k-DPC properties are characterized. At the same time, we obtain the relevant results of the unit interval graph G: for any vertex set W and any edge set F, G-W passing through F has the semi-paired 1-DPC property if and only if G is (2+r)-connected, where |W|=p, |F|=q, p+q≤r; G-W passing through F has the semi-paired k-DPC property if and only if G is (2k+r-1)-connected, where k≥2. It is shown that the existence of disjoint path covers in graphs is closely related to vertex connectivity and Hamiltonian properties. The research methods and results provide a theoretical basis for further study of the path coverage of interval graphs and other related graphs.

Restricted burning connectivity of graphs

Ruiying XUE,Zongtian WEI,Meijuan ZHAI

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 91-99, 109. doi:10.6040/j.issn.1671-9352.0.2022.326

Abstract ( 228 )

HTML ( 2 )

PDF (784KB) ( 361 )

Save

Figures and Tables | References | Related Articles | Metrics

Connectivity is an important indicator to measure the invulnerability of a network. This parameter is generalized from the perspective of graph burning, and the concept of restricted burning connectivity of graphs is proposed. On the basis of giving some basic graphs' restricted burning connectivity, the restricted burning connectivity calculation problems of the Cartesian product graph of paths and the spider graphs are studied by the mathematical programming method. By analyzing the relationship between restricted burning connectivity and graph structures, the advantages of this parameter in characterizing the invulnerability of networks are clarified.

Relative Gorenstein projective objects in comma categories

Jinping LUO,Li LIANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 100-104, 119. doi:10.6040/j.issn.1671-9352.0.2022.663

Abstract ( 198 )

HTML ( 1 )

PDF (639KB) ( 259 )

Save

References | Related Articles | Metrics

Given Abelian categories $\mathfrak{A}$ and $\mathfrak{B}$, and a right exact functor $\mathscr{F}$: $\mathfrak{A}$→$\mathfrak{B}$, an equivalent characterization for relative Gorenstein projective objects in the comma category ($\mathscr{F}$, $\mathfrak{B}$) is given in terms of relative Gorenstein projective objects in $\mathfrak{A}$ and $\mathfrak{B}$.

Stability function and torsion theory on exact categories

Zheng XIN,Dingguo WANG,Tiwei ZHAO

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 105-109. doi:10.6040/j.issn.1671-9352.0.2022.512

Abstract ( 190 )

HTML ( 1 )

PDF (657KB) ( 283 )

Save

References | Related Articles | Metrics

Stability function on exact categories is studied, and torsion classes and torsion-free classes are constructed by using the stability functions. Then, a class of torsion pairs is obtained.

Existence and approximate controllability of mild solutions for a class of Riemann-Liouville fractional evolution equations

Yuxin FENG,He YANG

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 110-119. doi:10.6040/j.issn.1671-9352.0.2022.592

Abstract ( 227 )

HTML ( 3 )

PDF (669KB) ( 340 )

Save

References | Related Articles | Metrics

The existence of mild solutions for a class of Riemann-Liouville fractional semilinear evolution equations in Banach space is proved by utilizing the cosine family theory of linear operators and Schauder′s fixed point theorem. The approximate controllability result is also established for the related control systems. An example is given to illustrate the application of abstract conclusions in the end.

Local commutant and wandering r-tuple of operators for unitary systems

Yujie ZHENG,Aifang LIU

JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE). 2024, 59(2): 120-126. doi:10.6040/j.issn.1671-9352.0.2022.487

Abstract ( 176 )

HTML ( 2 )

PDF (654KB) ( 123 )

Save

References | Related Articles | Metrics

According to the theory of completely wandering vector, the concepts of complete wandering operators with multiplicity r for unitary systems and local commutant are introduced. Then the properties of local commutant are given. Meanwhile, a characterization and other algebraic properties of the set of r-tuple complete wandering operators for unitary systems are obtained by means of the local commutators. Finally, some examples are given to illustrate the main results.

Table of Content