期刊：
LINEAR & MULTILINEAR ALGEBRA,2022年70(10):1854-1870 ISSN：0308-1087
通讯作者：
Li, Shuchao
作者机构：
[Xu, Baogen; Cao, Yanhua] East China Jiaotong Univ, Sch Sci, Nanchang, Jiangxi, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Mean first passage time;Kirchhoff index;spanning tree;n-prism network
期刊：
LINEAR & MULTILINEAR ALGEBRA,2022年70(11):2127-2149 ISSN：0308-1087
通讯作者：
Li, Shuchao
作者机构：
[Zhang, Huihui] Luoyang Normal Univ, Dept Math, Luoyang, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan, Peoples R China.
期刊：
LINEAR & MULTILINEAR ALGEBRA,2022年70(9):1768-1787 ISSN：0308-1087
通讯作者：
Li, Shuchao
作者机构：
[Yang, Ting; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
05C12;05C35;characteristic polynomial;Complex unit gain graph;cyclomatic number;matching number;rank
摘要：
Let [Math Processing Error] Gϕ be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of [Math Processing Error] Gϕ , written as [Math Processing Error] r(Gϕ) , is the rank of its adjacency matrix and denote by [Math Processing Error] α′(G) the matching number of the underlying graph G. In this contribution, based on combinatorial interpretation of all the coefficients of the characteristic polynomial of [Math Processing Error] Gϕ , we determine sharp upper and lower bounds on [Math Processing Error] r(Gϕ)−2α′(G). Furthermore, we establish sharp lower bounds on [Math Processing Error] r(Gϕ)−α′(G) and [Math Processing Error] r(Gϕ)/α′(G) . All the corresponding extremal complex unit gain graphs are characterized.
作者机构：
[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Li, Xuechao] Univ Georgia, Div Acad Enhancement, Athens, GA 30602 USA.;[Wei, Bing] Univ Mississippi, Dept Math, University, MS 38677 USA.
通讯机构：
[Wei, Bing] U;Univ Mississippi, Dept Math, University, MS 38677 USA.
关键词：
Cycle;Circumference;Specified vertices
摘要：
Let sigma(k)(G) = min {Sigma(k)(i=1) d(G)(v(i)) : {v(1), v(2),..., v(k)}is an independent set of G} for a graph G and an integer k >= 1. In 1991, Egawa et al. (1991) generalized a famous result on the existence of a hamiltonian cycle given by Ore (1960). In this paper, we will enhance the result of Egawa et al.. by providing the following result: Let G be a k-connected graph (k >= 2) and let S be an any vertex set with |S| <= k. Then G has a cycle of length at least min {|V(G)|, 2/k sigma(k)(G)} passing through S. (C) 2020 Elsevier B.V. All rights reserved.
期刊：
LINEAR & MULTILINEAR ALGEBRA,2021年69(13):2469-2490 ISSN：0308-1087
通讯作者：
Li, Shuchao
作者机构：
[Wei, Wei; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.;[Ma, Hongping] Jiangsu Normal Univ, Sch Math & Stat, Xuzhou, Jiangsu, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Hubei, Peoples R China.
期刊：
Advances in Applied Mathematics,2021年127:102164 ISSN：0196-8858
通讯作者：
Li, Shuchao(lscmath@mail.ccnu.edu.cn)
作者机构：
[Yu, Yuantian; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Shuchao Li] F;Faculty of Mathematics and Statistics, Central China Normal University, Wuhan 430079, PR China
关键词：
Wiener index;Closed walk;Adjacency polynomial;Laplacian polynomial;Edge cover polynomial;Independence polynomial
摘要：
This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikv?ri (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikv?ri (2010) [4] and Csikv?ri (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and 8, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length $ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikv & aacute;ri (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikv & aacute;ri (2010) [4] and Csikv & aacute;ri (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and 8, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length $ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having the maximum coefficients in absolute value among the trees of order n which are neither a path nor a star. (c) 2021 Elsevier Inc. All rights reserved.
期刊：
Journal of Algebraic Combinatorics,2021年53(4):921-943 ISSN：0925-9899
通讯作者：
Li, Shuchao
作者机构：
[Huang, Jing] South China Univ Technol, Sch Math, Guangzhou 510641, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Integral Cayley graph;Generalized dihedral group;Character;Irreducible representation
摘要：
A graph is said to be integral (resp. distance integral) if all the eigenvalues of its adjacency matrix (resp. distance matrix) are integers. Let H be a finite abelian group, and let
$${\mathscr {H}}=\langle H,b\,|\,b^2=1,bhb=h^{-1},h\in H\rangle $$
be the generalized dihedral group of H. The contribution of this paper is threefold. Firstly, based on the representation theory of finite groups, we obtain a necessary and sufficient condition for a Cayley graph over
$${\mathscr {H}}$$
to be integral, which naturally contains the main results obtained in Lu et al. (J Algebr Comb 47:585–601, 2018). Secondly, a closed-form decomposition formula for the distance matrix of Cayley graphs over any finite groups is derived. As applications, a necessary and sufficient condition for the distance integrality of Cayley graphs over
$${\mathscr {H}}$$
is displayed. Some simple sufficient (or necessary) conditions for the integrality and distance integrality of Cayley graph are exhibited, respectively, from which several infinite families of integral and distance integral Cayley graphs over
$${\mathscr {H}}$$
are constructed. And lastly, some necessary and sufficient conditions for the equivalence of integrity and distance integrity of Cayley graphs over generalized dihedral groups are obtained.
期刊：
MATCH-COMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY,2021年85(1):161-180 ISSN：0340-6253
通讯作者：
Li, Shuchao
作者机构：
[Deng, Kecai] Huaqiao Univ, Sch Math Sci, Quanzhou 362000, Fujian, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
期刊：
Applied Mathematics and Computation,2021年390:125598 ISSN：0096-3003
通讯作者：
Li, Shuchao
作者机构：
[Deng, Kecai] Huaqiao Univ, Sch Math Sci, Quanzhou 362000, Fujian, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
摘要：
For a given graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between n(u)(e) and n(v)(e) over all edges e = uv of G, where n(u)(e) and n(v)(e) are, respectively, the number of vertices of G lying closer to u than to v and the number of vertices of G lying closer to v than to u. The degree sequence of a tree is the sequence of the degrees (in descending order) of the non-leaf vertices. This paper determines those trees with a given degree sequence which have the greatest Mostar index. Consequently, all extremal trees with the greatest Mostar index are obtained in the sets of all trees of order n with the maximum degree, the number of leaves, the independence number and the matching number, respectively. On the other hand, some properties of trees with a given degree sequence which have the least Mostar index are given. We also determine those trees with exactly n - 3 or n - 4 leaves when the degree sequence is given, which have the least Mostar index. At last some numerical results are discussed, in which we calculate the Mostar indices of two sets of molecular graphs: octane isomers and benzenoid hydrocarbons; We compare their Mostar indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Mostar index is better than or as good as the models corresponding to the other distance-based indices. (C) 2020 Elsevier Inc. All rights reserved.
作者机构：
[Zhang, Huihui] Luoyang Normal Univ, Dept Math, Luoyang 471934, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Hitting time;Cover cost;Reverse cover cost;Diameter;Matching number
摘要：
For a connected graph G = (V-G, E-G), the cover cost of a vertex u in G is defined as CCG(u) = Sigma(v is an element of VG) H-uv and the reverse cover cost of a vertex v in G is defined as RCG(v) = Sigma(u is an element of VG) H-uv, where H-uv is the expected hitting time for random walk beginning at u to visit v. These two parameters were introduced as tractable variants of cover time on a graph. In this paper, some extremal problems on the (reverse) cover cost of a vertex in a tree with given graph parameters are considered. Firstly, the sharp lower bound on the cover cost among all n-vertex trees with given diameter (resp. matching number, domination number, vertex bipartition) is established. Secondly, the sharp lower bound on the reverse cover cost among all n-vertex trees with given diameter (resp. matching number, domination number) is established. All the corresponding extremal graphs are identified, respectively. At last, the unique n-vertex tree with given number of leaves having the minimum cover cost (resp. reverse cover cost) is characterized. (C) 2020 Elsevier B.V. All rights reserved.
期刊：
International Journal of Quantum Chemistry,2021年121(9):e26602- ISSN：0020-7608
通讯作者：
Li, Shuchao
作者机构：
[Deng, Kecai] Huaqiao Univ, Sch Math Sci, Quanzhou, Peoples R China.;[Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
摘要：
<jats:title>Abstract</jats:title><jats:p>Given a graph <jats:styled-content><jats:italic>G</jats:italic></jats:styled-content>, the Mostar index <jats:styled-content><jats:italic>Mo</jats:italic>(<jats:italic>G</jats:italic>)</jats:styled-content> is the sum of absolute values of the differences between <jats:styled-content><jats:italic>n</jats:italic><jats:sub><jats:italic>u</jats:italic></jats:sub>(<jats:italic>e</jats:italic>)</jats:styled-content> and <jats:styled-content><jats:italic>n</jats:italic><jats:sub><jats:italic>v</jats:italic></jats:sub>(<jats:italic>e</jats:italic>)</jats:styled-content> over all edges <jats:styled-content><jats:italic>e</jats:italic> = <jats:italic>uv</jats:italic></jats:styled-content> of <jats:styled-content><jats:italic>G</jats:italic></jats:styled-content>, where <jats:styled-content><jats:italic>n</jats:italic><jats:sub><jats:italic>u</jats:italic></jats:sub>(<jats:italic>e</jats:italic>)</jats:styled-content> and <jats:styled-content><jats:italic>n</jats:italic><jats:sub><jats:italic>v</jats:italic></jats:sub>(<jats:italic>e</jats:italic>)</jats:styled-content> are, respectively, the number of vertices of <jats:styled-content><jats:italic>G</jats:italic></jats:styled-content> lying closer to <jats:styled-content><jats:italic>u</jats:italic></jats:styled-content> than to <jats:styled-content><jats:italic>v</jats:italic></jats:styled-content> and the number of vertices of <jats:styled-content><jats:italic>G</jats:italic></jats:styled-content> lying closer to <jats:styled-content><jats:italic>v</jats:italic></jats:styled-content> than to <jats:styled-content><jats:italic>u</jats:italic></jats:styled-content>. A tree‐like polyphenyl is a polycyclic aromatic hydrocarbon consisting of benzene rings, whose chemical graph will tend to a tree after contracting each hexagon into a vertex (the tree is called a contracted tree). A polyphenyl chain is a tree‐like polyphenyl whose contracted tree is a path. In this paper, those polyphenyl chains with <jats:styled-content><jats:italic>n</jats:italic></jats:styled-content> benzene rings having the least, the second least, the greatest and the second greatest Mostar indices are determined, respectively. Those tree‐like polyphenyls with <jats:styled-content><jats:italic>n</jats:italic></jats:styled-content> benzene rings having the least and the second least Mostar indices are also identified. What is more, some properties of those tree‐like polyphenyls with <jats:styled-content><jats:italic>n</jats:italic></jats:styled-content> benzene rings having the greatest Mostar index are obtained. At the end we state some further research problems.</jats:p>
作者机构：
[Wei, Wei; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Mixed graphs;Complex unit gain graph;A(alpha)-matrix;Multiplicity;Switching equivalent
摘要：
The work of Wang et al. (2020) established an upper bound on the multiplicity of a real number as an adjacency eigenvalue of an undirected simple graph G according to the dimension of its cycle space and the number of its pendants. The work of Cardoso et al. (2018) studied the multiplicity of alpha as an eigenvalue of alpha D(G) + (1 - alpha)A(G), alpha is an element of [0, 1), where D(G) is the diagonal matrix of degrees and A(G) is the adjacency matrix of G, which was ported to signed graphs by the work of Belardo et al. (2019). Here, on the one hand, we consider both the Wang-Wei-Jin-type and Cardoso-Pasten-Rojo-type routines developed for graphs to the level of mixed graphs and complex unit gain graphs. On the other hand, we consider Belardo-Brunetti-Ciampella-type routines developed for signed graphs to the level of mixed graphs and complex unit gain graphs. We show that, with suitable adaption, such routines can be successfully ported to mixed graphs and complex unit gain graphs. By these obtained results in the current paper, the corresponding results for undirected graphs, oriented graphs and signed graphs are deduced. (C) 2020 Elsevier B.V. All rights reserved.
作者机构：
[Zhang, Leilei; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.;[Li, Qishun] Beijing Jiaotong Univ, Sch Elect Engn, Beijing 100044, Peoples R China.;[Zhang, Minjie] Hubei Univ Arts & Sci, Sch Math & Stat, Xiangyang 441053, Peoples R China.
通讯机构：
[Zhang, Minjie] H;Hubei Univ Arts & Sci, Sch Math & Stat, Xiangyang 441053, Peoples R China.
关键词：
Random polyphenylene chain;Schultz index;Gutman index;Multiplicative degree-Kirchhoff;Additive degree-Kirchhoff index
摘要：
This article is devoted to establish the explicit analytical expressions for the expected values of the Schultz index, Gutman index, multiplicative degree-Kirchhoff index and additive degree-Kirchhoff index of a random polyphenylene chain. The average values of these four indices with respect to the set of all polyphenylene chains with n hexagons are obtained. (C) 2019 Elsevier B.V. All rights reserved.
摘要：
The resistance between two nodes in some electronic networks has been studied extensively. Let G(n) be a generalized phenylene with n 6-cycles and n 4-cycles. Using series and parallel rules and the Delta - Y transformations we obtain explicit formulae for the resistance distance between any two points of G(n). To the best of our knowledge {G(n)}(n=1)(infinity) is a nontrivial family with diameter going to infinity for which all resistance distances have been explicitly calculated. We also determine the maximal resistance distance and the minimal resistance distance in G(n). The monotonicity and some asymptotic properties of resistance distances in G(n) are given. At last some numerical results are discussed, in which we calculate the Kirchhoff indices of a set of benzenoid hydrocarbons; We compare their Kirchhoff indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Kirchhoff index is better than or as good as the models corresponding to the other distance-based indices.
期刊：
Linear Algebra and its Applications,2020年588:19-53 ISSN：0024-3795
通讯作者：
Li, Shuchao
作者机构：
[Wei, Wei; Li, Shuchao; Feng, Zhimin] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Mixed graph;Hermitian adjacency matrix;Inertia index
摘要：
Given a graph G, the mixed graph D-G is obtained from G by orienting some of its edges, where G is called the underlying graph of D-G. Let p(D-G), n(D-G) (resp. P(G), n(G)) be the positive inertia index and negative inertia index of D-G (resp. G). In this paper, we first establish the inequalities -d(G) p(D-G)-p(G) <= d(G) and d(G) <= n(D-G) - n(G) <= d(G), where d(G) is the dimension of cycle space of G. Furthermore, all the corresponding extremal graphs are characterized. (C) 2019 Elsevier Inc. All rights reserved.
摘要：
<jats:title>Abstract</jats:title><jats:p>It is well known to us that a graph of diameter has at least eigenvalues. A graph is said to be <jats:italic>Laplacian</jats:italic> (resp, <jats:italic>normalized Laplacian</jats:italic>) ‐<jats:italic>extremal</jats:italic> if it is of diameter having exactly distinct Laplacian (resp, normalized Laplacian) eigenvalues. A graph is split if its vertex set can be partitioned into a clique and a stable set. Each split graph is of diameter at most 3. In this paper, we completely classify the connected bidegreed Laplacian (resp, normalized Laplacian) 2‐extremal (resp, 3‐extremal) split graphs using the association of split graphs with combinatorial designs. Furthermore, we identify connected bidegreed split graphs of diameter 2 having just four Laplacian (resp, normalized Laplacian) eigenvalues.</jats:p>
作者机构：
[He, Xiaocong; Wei, Wei; Li, Shuchao] Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
通讯机构：
[Li, Shuchao] C;Cent China Normal Univ, Fac Math & Stat, Wuhan 430079, Peoples R China.
关键词：
Diameter;Spectral radius;The eccentricity matrix;The least eigenvalue
摘要：
The eccentricity matrix epsilon(G) of a graph G is constructed from the distance matrix of G by keeping only the largest distances for each row and each column. This matrix can be interpreted as the opposite of the adjacency matrix obtained from the distance matrix by keeping only the distances equal to 1 for each row and each column. In this paper we focus on the eccentricity matrix of graphs. Let T be an n-vertex tree and let epsilon(n)(T) be the least epsilon-eigenvalue of T. On the one hand, we determine the n-vertex trees with the minimum epsilon-spectral radius. On the other hand, for n >= 3, we show that epsilon(n)(T) <= -2 with equality if and only if T is a star. As a consequence, we solve two conjectures proposed by Wang et al. (2018). Furthermore, we identify all the trees with given order and diameter having the minimum epsilon-spectral radius. Finally, we determine all the n-vertex connected graphs whose maximum degrees are less than n - 1 and least epsilon-eigenvalues are in [-2 root 2, -2]. (C) 2020 Elsevier B.V. All rights reserved.
