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June 2001 - present
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Publications (245)
Let G be a graph and let Pn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_n$$\end{document} be a path on n vertices. A spanning subgraph H of G is called a {P3,P4,P5...
The eccentricity matrix of a graph is obtained from the distance matrix by keeping the
entries that are largest in their row or column, and replacing the remaining entries by zero. This matrix can be interpreted as an opposite to the adjacency matrix, which is on the contrary obtained from the distance matrix by keeping only the entries equal to 1....
A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$ can be extended to a perfect matching of $G$, where $k$ is a positive integer. We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a perfect matching excluding $e$. In this paper, we first establish a lower bound on the size (resp. the spectral radius)...
Let λ2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda _2$$\end{document} be the second largest eigenvalue of the adjacency matrix of a connected graph. In 2021,...
The Steiner k-eccentricity of a vertex in a graph G is the maximum Steiner distance over all k-subsets containing the vertex. The average Steiner k-eccentricity of G is the mean value of all vertices’ Steiner k-eccentricities in G. Let Tn be the set of all n-vertex trees, Tn,Δ be the set of n-vertex trees with maximum degree Δ, Tn,Δk be the set of...
Let G be a graph and let k⩾2 be an integer. A {K1,j:1⩽j⩽k}-factor of G is a spanning subgraph of G, in which each component is isomorphic to a member in {K1,j:1⩽j⩽k}. In this paper, we first establish a lower bound on the size (resp. the spectral radius) of G to guarantee that G contains a {K1,j:1⩽j⩽k}-factor. Then we determine an upper bound on th...
We first establish a lower bound on the size and spectral radius of a graph G to guarantee that G contains a fractional perfect matching. Then, we determine an upper bound on the distance spectral radius of a graph G to ensure that G has a fractional perfect matching. Furthermore, we construct some extremal graphs to show all the bounds are best po...
For k⩾2, a P⩾k-factor of a graph G is a spanning subgraph F of G such that each component of F is a path with at least k vertices. A graph G is a P⩾k-factor covered graph if for each edge e in E(G), there exists a P⩾k-factor containing the edge e. Let Q(G) and D(G) be the signless Laplacian matrix and the distance matrix of a graph G, respectively....
An independent set in a graph G is a set of pairwise non-adjacent vertices of G. The independence number, α, of G is the maximum cardinality of an independent set in G. An independent set in G is maximum if it has cardinality α. Mohr and Rautenbach determined the n-vertex trees (resp. disconnected graphs) with independence number α having the large...
The Aα matrix of a graph G is defined by Aα(G)=αD(G)+(1−α)A(G),0≤α≤1, where D(G) is the diagonal matrix of degrees and A(G) is the adjacency matrix of G. The Aα-spectrum of a graph G, denoted by SpecAα(G), is the set of eigenvalues together with their multiplicities of Aα(G). A graph G is said to be determined by the generalized Aα-spectrum (DGAαS...
Let G be an n-vertex graph. A matching in G is a set of independent edges, i.e., no two edges in the set are adjacent in G. The matching number is the maximal size of a matching in G. Nikiforov (Appl Anal Discrete Math 11(1):81–107, 2017) proposed the \(A_{\alpha }\)-matrix of a graph G, as follows:
$$\begin{aligned} A_{\alpha }(G)=\alpha D(G)+(1-...
The eccentricity matrix E(G) of a graph G is derived from the corresponding distance matrix by keeping only the largest non-zero elements for each row and each column and leaving zeros for the remaining ones. The E-eigenvalues of a graph G are those of its eccentricity matrix, in which the maximum modulus is called the E-spectral radius. In this pa...
The L-index (resp. Q-index) of a graph G is the largest eigenvalue of the Laplacian matrix (resp. signless Laplacian matrix) of G. Very recently, Lou, Guo and Wang [10] determined the graph with fixed size and diameter having the maximum Q-index (resp. L-index). As a continuance of their result, in this paper we order all the graphs with given size...
Let $\mathcal{F}$ denote a set of graphs. A graph $G$ is said to be $\mathcal{F}$-free if it does not contain any element of $\mathcal{F}$ as a subgraph. The Tur\'an number is the maximum possible number of edges in an $\mathcal{F}$-free graph with $n$ vertices. It is well known that classical Tur\'an type extremal problem aims to study the Tur\'an...
This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind (N-matrix for short) introduced by Mohar [25]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from u to v is equal to the sixth root of unity ω=1+i32 (and its symmetric entry i...
Given a connected graph G with vertex set VG, the hitting time HG(u,v) of vertices u and v in G is defined as the expected number of steps that a random walk takes to go from u to v. Then the ZZ index of G, denoted by ψ(G), is defined as ψ(G)=max{u,v}⊆VGHG(u,v). This hitting-time-based invariant was first proposed by Zhu and Zhang (2021). In this p...
Given a graph $G$, the adjacency matrix and degree diagonal matrix of $G$ are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} proposed the $A_{\alpha}$-matrix: $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G),$ where $\alpha\in [0, 1]$. The largest eigenvalue of this novel matrix is called the $A_\alpha$-index of $G$. In this pa...
Based on elementary geometry, Gutman proposed a novel graph invariants called the Sombor index SO(G), which is defined as SO(G)=∑uv∈E(G)dG2(u)+dG2(v), where dG(u) and dG(v) denote the degree of u and v in G, respectively. It has been proved that the Sombor index could predict some physicochemical properties. In this paper, we characterize the extre...
The eccentricity matrix E(G) of a graph G is derived from the corresponding distance matrix by keeping only the largest non-zero elements for each row and each column and leaving zeros for the remaining ones. The E-eigenvalues of a graph G are those of its eccentricity matrix. The E-spectrum of G is a multiset consisting of its distinct E-eigenvalu...
It is well known that spectral Turán type problem is one of the most classical problems in graph theory. In this paper, we consider the spectral Turán type problem. Let G be a graph and let G be a set of graphs, we say G is G-free if G does not contain any element of G as a subgraph. Denote by λ1 and λ2 the largest and the second largest eigenvalue...
Let Fn be the fan graph with n⩾3 vertices. The distance d(i,j) between any two distinct vertices i and j of Fn is the length of the shortest path connecting i and j. Let D^ be the n×n symmetric matrix with diagonal entries equal to zero and off-diagonal entries equal to d(i,j). In this paper, we find two positive semidefinite matrices Lo and Le suc...
A P⩾k-factor (k⩾2) of a graph G is a spanning subgraph of G in which each component is a path of order at least k. A graph G is called a P⩾k-factor covered graph if for each edge e of G, there is a P⩾k-factor covering e. In this paper, we first establish two lower bounds on the size of a graph G, in which one bound guarantees that G contains a P⩾2-...
Let G be a graph with vertex set VG={v1,v2,…,vn} and edge set EG, and let di be the degree of the vertex vi. The ABC matrix of G has the value (di+dj−2)/(didj) if vivj∈EG, and 0 otherwise, as its (i,j)-entry. Let γ1,γ2,…,γn be the eigenvalues of the ABC matrix of G in a non-increasing order. Then the ABC Estrada index of G is defined as EEABC(G)=∑i...
Given a graph G and an edge \(e=xy\) in G, let \(n_x(e)\) and \(n_y(e)\) be the number of vertices that have the distance to x less than that to y, and the number of vertices that have the distance to y less than that to x, respectively. The contribution of e is defined as \(|n_x(e)-n_y(e)|\). The Mostar index of G is the sum of all edge contributi...
It is well known that spectral Tur\'{a}n type problem is one of the most classical {problems} in graph theory. In this paper, we consider the spectral Tur\'{a}n type problem. Let $G$ be a graph and let $\mathcal{G}$ be a set of graphs, we say $G$ is \textit{$\mathcal{G}$-free} if $G$ does not contain any element of $\mathcal{G}$ as a subgraph. Deno...
Let Tn,k be the set of k-ary trees of order n=(k−1)t+2, where k-ary trees are trees in which every vertex has degree 1 or k. For a connected graph G=(V(G),E(G)), the cover cost (resp. reverse cover cost) of a vertex u in G is defined as CCG(u)=∑v∈V(G)Huv (resp. RCG(u)=∑v∈V(G)Hvu), where Huv is the expected hitting time for random walk beginning at...
Let G be a graph on n vertices, its adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017, Nikiforov [20] introduced the matrix Aα(G)=αD(G)+(1−α)A(G) for α∈[0,1]. The Aα-spectrum of a graph G consists of all the eigenvalues (including the multiplicities) of Aα(G). A graph G is said to be determined by the g...
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. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertic...
Let A(G) and D(G) be the adjacency matrix and the degree diagonal matrix of a graph G, respectively. For any real α∈[0,1], Nikiforov [Appl Anal Discrete Math. 2017;11:81–107] proposed the matrix Aα(G)=αD(G)+(1−α)A(G), whose largest eigenvalue is called the Aα-index of G. In this contribution, we study this novel and interesting matrix. On the one h...
Given a simple graph G, a (c,s)-cluster of G is a pair of vertex subsets (C,S), where size of C is c (≥ 2) and every vertex in C shares the same set S of s neighbours. Let MG be a mixed graph whose underlying graph G contains a (c,s)-cluster (C,S) and let MH be a mixed graph on c vertices. Then MG(H) is a graph obtained from MG by adding edges betw...
Let σk(G)=min{∑i=1kdG(vi):{v1,v2,…,vk}is an independent set ofG} 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...
A graph is said to be distance-integral if every eigenvalue of its distance matrix is an integer. In this paper, we study the distance spectrum of abelian Cayley graphs and a class of non-abelian Cayley graphs, namely Cayley graphs over the dicyclic group \(T_{4n}=\langle a,b\,|\,a^{2n}=1, a^n=b^2, b^{-1}ab=a^{-1}\rangle \) of order 4n. Based on th...
Let $G$ be a graph on $n$ vertices, its adjacency matrix and degree diagonal matrix are denoted by $A(G)$ and $D(G)$, respectively. In 2017, Nikiforov \cite{0007} introduced the matrix $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ for $\alpha\in [0, 1].$ The $A_\alpha$-spectrum of a graph $G$ consists of all the eigenvalues (including the multipliciti...
Let G be a connected graph with vertex set VG. The eccentric resistance-distance sum of G is defined as ξR(G)=∑{u,v}⊆VG(εG(u)+εG(v))Ruv, where εG(⋅) is the eccentricity of the corresponding vertex and Ruv is the resistance-distance between u and v in G. In this paper, among the bipartite graphs of diameter 2, the graphs having the smallest and the...
This contribution gives an extensive study on spectra of mixed graphs via its Hermitian adjacency matrix of the second kind introduced by Mohar [21]. This matrix is indexed by the vertices of the mixed graph, and the entry corresponding to an arc from $u$ to $v$ is equal to the sixth root of unity $\omega=\frac{1+{\bf i}\sqrt{3}}{2}$ (and its symme...
For a given graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between nu(e) and nv(e) over all edges e=uv of G, where nu(e) and nv(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...
For a connected graph G=(VG,EG), the cover cost of a vertex u in G is defined as CCG(u)=∑v∈VGHuv and the reverse cover cost of a vertex v in G is defined as RCG(v)=∑u∈VGHuv, where Huv 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 p...
Given a graph G, the Mostar index Mo(G) is the sum of absolute values of the differences between nu(e) and nv(e) over all edges e = uv of G, where nu(e) and nv(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. A tree‐like polyphenyl is a polycyclic aromatic hyd...
Let G be a graph, whose adjacency matrix and degree diagonal matrix are denoted by A(G) and D(G), respectively. In 2017 Nikiforov (2017) merged the A- and Q-spectral theories to proposed the Aα matrix: Aα(G)=αD(G)+(1−α)A(G), α∈[0,1]. Its largest eigenvalue is called the Aα-index of G. In this paper, we focus on specifying the property of the Aα mat...
The eccentricity matrix E ðGÞ of a graph G is derived from the distance matrix by keeping for each row and each column only the largest distances and leaving zeros in the remaining ones. The E-eigenvalues of a graph G are those of its eccentricity matrix E ðGÞ. The E-spectrum of the graph G is the multiset of its E-eigenvalues, where the maximum mo...
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 knowled...
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 α as an eigenvalue of αD(G)+(1−α)A(G),α∈[0,1), where D(G...
The eccentricity matrix ε(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 fo...
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 vertex u than to vertex v and the number of vertices of G lying closer to vertex v than to v...
It is well known to us that a graph of diameter l has at least l + 1 eigenvalues. A graph is said to be Laplacian (resp, normalized Laplacian) l‐extremal if it is of diameter l having exactly l + 1 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...
Given a connected graph G = ( V G , E G ) with x , y ∈ V G , the hitting time H G ( x , y ) is defined as the expected number of steps that a simple random walk takes to go from x to y. A hitting-time-based invariant, called the ZZ index, was first proposed by Zhu and Zhang [The hitting time of random walk on unicyclic graphs. Linear Multilinear Al...
The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let Ln be a linear hexagonal chain with n 6-cycles. Then identifying the opposite lateral edges of Ln in an ordered way yields the linear hexagonal cylinder chain, written as Rn. We obtain explicit formulae for the resistance di...
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...
Let Gϕ be an n-vertex complex unit gain graph and let G be its underlying graph. The adjacency rank of Gϕ, written as r(Gϕ), is the rank of its adjacency matrix and denote by 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 Gϕ...
In this paper, the Laplacian matrix of an n-prism network Ugn and its applications are studied. Firstly, the relation between the Laplacian matrix of Ugn and that of U0n is established. Secondly, the analytical expression for the product of all the nonzero Laplacian eigenvalues of Ugn is obtained. At the same time, the sum of the reciprocals of all...
A segment of a tree T is a path whose end vertices have degree 1 or at least 3, while all internal vertices have degree 2. The lengths of all the segments of T form its segment sequence, in analogy to the degree sequence. For a connected graph G=(V(G),E(G)), the cover cost (resp. reverse cover cost) of a vertex u in G is defined as CCG(u)=∑v∈V(G)Hu...
A path P is a segment of a tree if the endpoints of P are of degree 1 or at least 3, and each of the rest vertices are of degree 2 in the tree. The lengths of all the segments of this tree form its segment sequence. Denote by Tl the set of all trees on n vertices with the segment sequence l=(l1,l2,…,lm), where l1⩾l2⩾⋯⩾lm. In this paper, the extrema...
Given a simple graph G, let \(Z(G),\, p(G),\, \Phi (G), ex(G)\) and M(G), respectively, be the zero forcing number, the number of pendant vertices, the cyclomatic number, the number of exterior major vertices and the maximum nullity of G. Wang et al. (Linear Multilinear Algebra, 2018. https://doi.org/10.1080/03081087.2018.1545829) established upper...
Let G be a connected graph with order n, maximum degree Δ⩾2 and nullity η(G). Zhou, Wong and Sun [6] conjectured that η(G)⩽(Δ−2)n+2Δ−1 with equality if and only if G is a cycle Cn with n divisible by 4 or a complete bipartite graph Kn2,n2. Very recently, Cheng, Liu and Liu [2] confirmed this conjecture. In this paper, we give a much shorter proof f...
Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[...
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 a...
Given a graph G, the mixed graph DG is obtained from G by orienting some of its edges, where G is called the underlying graph of DG. Let p(DG), n(DG) (resp. p(G), n(G)) be the positive inertia index and negative inertia index of DG (resp. G). In this paper, we first establish the inequalities −d(G)⩽p(DG)−p(G)⩽d(G) and −d(G)⩽n(DG)−n(G)⩽d(G), where d...
Given an n-vertex graph G with maximum degree Δ, the mixed graph DG is constructed from G by orienting some of its edges, where G is called the underlying graph of DG. Let rH(DG) be the H-rank of DG and let α(G) be the independence number of G. In this paper, we firstly determine the maximum nullity of n-vertex mixed graphs with maximum degree Δ. T...
A signed graph Γ(G) is a graph with a sign attached to each of its edges, where G is the underlying graph of Γ(G). The energy of a signed graph Γ(G) is the sum of the absolute values of the eigenvalues of the adjacency matrix A(Γ(G)) of Γ(G). The random signed graph model Gn(p,q) is defined as follows: Let p,q≥0 be fixed, 0<p+q<1. Given a set of n...
Let denote a molecular graph of linear [n] phenylene with n hexagons and n squares, and let the Möbius phenylene chain be the graph obtained from the by identifying the opposite lateral edges in reversed way. Utilizing the decomposition theorem of the normalized Laplacian characteristic polynomial, we study the normalized Laplacian spectrum of , wh...
The first Zagreb index M1 of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index M2 of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices. In Furtula et al. (2014) Furtula, Gutman and Ediz first proposed the reduced second Zagreb index as RM2(G)=∑uv∈E(G)(d(u)−...
The first and the second Zagreb indices of a graph G are defined as \(M_1(G)= \sum _{v\in V_G}d_v^2 \) and \( M_2(G)= \sum _{uv\in E_G}d_ud_v\), where \(d_v,\, d_u\) are the degrees of vertices \(v,\, u\) in G. The difference of Zagreb indices of G is defined as \(\Delta M(G)=M_2(G)-M_1(G)\). A cactus is a connected graph in which every block is ei...
Given a simple graph G, let A(G) be its adjacency matrix and α′(G) be its matching number. The rank of G, written as r(G), refers to the rank of A(G). In this paper, some relations between the rank and the matching number of a graph are studied. Firstly, it is proved that −2d(G)⩽r(G)−2α ′ (G)⩽N o , where d(G) and N o are, respectively, the dimensio...
In this paper, the extremal problems on the phenylene chains with respect to some graph invariants are studied. All the graphs minimizing (resp. maximizing) the coefficients sum of the permanental polynomial, the spectral radius, the Hosoya index and the Merrifield–Simmons index among all the phenylene chains each of which contains n four-membered...
Given a connected graph G, the diamond hierarchical graph S(G) is formed by adding two new vertices ve,we for each edge e=uv in G and then deleting edge e and adding in edges uve,uwe and vev,wev. In this paper, the eigenvalues and eigenvectors of the probability transition matrix of a random walks on SG are completely provided at first. Then the ex...
Let DG˜ be an n-vertex weighted mixed graph with Hermitian-adjacency matrix H(DG˜). The characteristic polynomial of the weighted mixed graph DG˜ is defined asϕ(DG˜,λ)=det(λIn−H(DG˜))=∑r=0nαrλn−r and the H-rank of DG˜, written as rk(DG˜), is the rank of H(DG˜).
In this paper, we begin by interpreting all the coefficients of the characteristic poly...
The sum of distances between all pairs of vertices (denoted by σ(⋅) and called the Wiener index) and the number of subtrees (denoted by F(⋅) and called the subtree index) of a graph G are two representative graph invariants that have been extensively studied. The “local” version of these graph invariants (i.e. sum of distances from a given vertex,...
Let Ln denote the linear hexagonal chain containing n hexagons. Then identifying the opposite lateral edges of Ln in ordered way yields TUHC[2n, 2], the zigzag polyhex nanotube, whereas identifying those of Ln in reversed way yields Mn, the hexagonal Möbius chain. In this article, we first obtain the explicit formulae of the multiplicative degree‐K...
The resistance between two nodes in some resistor networks has been studied extensively by mathematicians and physicists. Let $L_n$ be a linear hexagonal chain with $n$\, 6-cycles. Then identifying the opposite lateral edges of $L_n$ in ordered way yields the linear hexagonal cylinder chain, written as $R_n$. We obtain explicit formulae for the res...
Given a simple graph G=(VG,EG) with vertex set VG and edge set EG, the mixed graph G˜ is obtained from G by orienting some of its edges. Let H(G˜) denote the Hermitian adjacency matrix of G˜ and A(G) be the adjacency matrix of G. The H-rank (resp. rank) of G˜ (resp. G), written as rk(G˜) (resp. r(G)), is the rank of H(G˜) (resp. A(G)). Denote by d(...
Let G be a connected graph. The first and the second largest distance signless Laplacian eigenvalues of G are denoted by (Formula presented.) and (Formula presented.). In this paper, we determine the graphs with the minimum (Formula presented.) among n-vertex graphs with given matching number. We also establish sharp lower bounds on (Formula presen...
Let G be a graph of order n and let di be the degree of the vertex vi in G for i=1,2,…,n. The weighted adjacency matrix Adb of G is defined so that its (i, j)-entry is equal to [Formula presented] if the vertices vi and vj are adjacent, and 0 otherwise. The spectral radius ϱ1 and the energy Edb of the Adb-matrix are examined. Lower and upper bounds...
A signed graph $\Gamma(G)$ is a graph with a sign attached to each of its edges, where $G$ is the underlying graph of $\Gamma(G)$. The energy of a signed graph $\Gamma(G)$ is the sum of the absolute values of the eigenvalues of the adjacency matrix $A(\Gamma(G))$ of $\Gamma(G)$. The random signed graph model $\mathcal{G}_n(p, q)$ is defined as foll...
Given a simple graph $G=(V_G, E_G)$ with vertex set $V_G$ and edge set $E_G$, the mixed graph $\widetilde{G}$ is obtained from $G$ by orienting some of its edges. Let $H(\widetilde{G})$ denote the Hermitian adjacency matrix of $\widetilde{G}$ and $A(G)$ be the adjacency matrix of $G$. The $H$-rank (resp. rank) of $\widetilde{G}$ (resp. $G$), writte...
Given a simple connected graph G, the k-triangle graph of G, written by Tk(G), is obtained from G by adding k new vertices ui1,ui2,…,uik for each edge ei=uv in G and then adding in edges uui1,uui2,…,uuik and ui1v,ui2v,…,uikv. In this paper, the eigenvalues and eigenvectors of the probability transition matrix of random walks on Tk(G) are completely...
Let G be a connected graph. The eccentric connectivity index ξc(G) of G is defined as [Formula presented], where the eccentricity [Formula presented]. Zhang et al. (2012) studied the minimal eccentric connectivity indices of graphs. As a continuance of it, in this paper we consider these problems on bipartite graphs. We obtain lower bounds on ξc(G)...
Let be the molecular graph of the linear [n]phenylene with n hexagons and n − 1 squares, and let be the graph obtained by attaching four‐membered rings to the terminal hexagons of . In this article, the normalized Laplacian spectrum of consisting of the eigenvalues of two symmetric tridiagonal matrices of order 3n is determined. An explicit closed‐...
Given a connected graph G, two types of graph transformations on G are considered. The graph is obtained by applying the first transformation on G, i.e. it is formed by adding a new triangle for each edge e=uv in G and then adding in edges and , whereas the graph is obtained by applying the second transformation on G, i.e. it is formed by adding a...
Given a graph G, the mixed graph is obtained from G by orienting some of its edges (G is also called the underlying graph of ). Let be the Hermitian energy of and let be the matching number of the underlying graph G. In this paper, we first establish the inequality and characterize all the mixed graphs which make hold. Furthermore, we obtain , wher...
Given a connected graph G, the edge-Szeged index Sze(G) is defined as Sze(G)=∑e=uv∈Emu(e)mv(e), where mu(e) and mv(e) are, respectively, the number of edges of G lying closer to vertex u than to vertex v and the number of edges of G lying closer to vertex v than to vertex u. In this paper, some extremal problems on the edge-Szeged index of unicycli...
Let Gw be an edge-weighted graph and let A(Gw) be its adjacency matrix. The positive inertia index (respectively, the negative inertia index) of Gw, denoted by pGw⁺ (resp. pGw⁻), is defined to be the number of positive eigenvalues (resp. negative eigenvalues) of A(Gw). Clearly, pGw⁺+pGw⁻ is the rank of Gw, whereas pGw⁺−pGw⁻ is defined as the signat...
Octagonal systems are tree-like graphs comprised of octagons that represent a class of polycyclic conjugated hydrocarbons. In this paper, a roll-attaching operation for the calculation of the characteristic polynomials of octagonal chain graphs is proposed. Based on these characteristic polynomials, the extremal octagonal chains with n octagons hav...
Given a connected graph G=(VG,EG), the eccentricity distance sum (EDS) of G is defined as ξd(G)=∑{u,v}⊆VG(ε(u)+ε(v))dG(u,v) and the eccentric connectivity index (ECI) of G is defined as ξc(G)=∑v∈VGε(v)dG(v), where ε(v),dG(v) and dG(u,v) are the eccentricity of v, the degree of v and the distance between u and v in G, respectively. In this paper, so...
An oriented graph $G^\sigma$ is a digraph without loops or multiple arcs whose underlying graph is $G$. Let $S\left(G^\sigma\right)$ be the skew-adjacency matrix of $G^\sigma$ and $\alpha(G)$ be the independence number of $G$. The rank of $S(G^\sigma)$ is called the skew-rank of $G^\sigma$, denoted by $sr(G^\sigma)$. Wong et al. [European J. Combin...
Tree-like octagonal systems are cata-condensed systems of octagons, which represent a class of polycyclic conjugated hydrocarbons. An octagonal chain is a cata-condensed octagonal system with no branchings. In this paper, the extremal octagonal chains with n octagons having the minimum and maximum coefficients sum of the permanental polynomial are...
Let be a simple graph with vertex set and edge set . By orienting a subset of , we get a mixed graph . Let be the Hermitian-adjacency matrix of , the rank of , written as , is called the H-rank of . Denote by the dimension of cycle spaces of G, that is , where denotes the number of connected components of G. Let be the independence number of G. In...
Let Wn be a linear pentagonal chain with 2n pentagons. In this article, according to the decomposition theorem for the normalized Laplacian polynomial of Wn, we obtain that the normalized Laplacian spectrum of Wn consists of the eigenvalues of two special matrices: LA of order 3n+1 andLS of order 2n+1. Together with the relationship between the roo...
An oriented graph Gσ is a digraph without loops and multiple arcs, where G is the underlying graph of Gσ. Let S(Gσ) denote the skew-adjacency matrix of Gσ, and A(G) be the adjacency matrix of G. The rank (resp. skew-rank) of G (resp. Gσ), written as r(G) (resp. sr(Gσ)), refers to the rank of A(G) (resp. S(Gσ)). It is natural and interesting to stud...
A connected graph is said to be a cactus if each of its blocks is either a cycle or an edge. Let Cn be the set of all n-vertex cacti with circumference at least 4, and let Cn,k be the set of all n-vertex cacti containing exactly k⩾1 cycles where n⩾3k+1. In this paper, lower bounds on the difference between the (revised) Szeged index and Wiener inde...
The eccentric distance sum (EDS) of a connected graph G is defined as ξd(G)=∑{u,v}⊆VG(εG(u)+εG(v))dG(u,v), where εG(⋅) is the eccentricity of the corresponding vertex and dG(u,v) is the distance between u and v in G. In this paper, some extremal problems on the EDS of an n-vertex graph with respect to other two graph parameters are studied. Firstly...