scholarly journals On Graphs whose Orientations are Determined by their Hermitian Spectra

10.37236/9640 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Yi Wang ◽  
Bo-Jun Yuan
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Simple Graph ◽  
Equivalence Classes ◽  
Cycle Space ◽  
Mixed Graph ◽  
Cycle Basis ◽  
Underlying Graph ◽  
Mixed Graphs ◽  
Special Cycle

A mixed graph $D$ is obtained from a simple graph $G$, the underlying graph of $D$, by orienting some edges of $G$. A simple graph $G$ is said to be ODHS (all orientations of $G$ are determined by their $H$-spectra) if every two $H$-cospectral graphs in $\mathcal{D}(G)$ are switching equivalent to each other, where $\mathcal{D}(G)$ is the set of all mixed graphs with $G$ as their underlying graph. In this paper, we characterize all bicyclic ODHS graphs and construct infinitely many ODHS graphs whose cycle spaces are of dimension $k$. For a  connected graph $G$ whose cycle space is of dimension $k$, we also obtain an achievable upper bound $2^{2k-1} + 2^{k-1}$ for the number of switching equivalence classes in $\mathcal{D}(G)$, which naturally is an upper bound of the number of  cospectral classes in $\mathcal{D}(G)$. To achieve these, we propose a valid method to estimate the number of switching equivalence classes in $\mathcal{D}(G)$ based on the strong cycle basis, a special cycle basis  introduced in this paper.

scholarly journals On the Spectra of General Random Mixed Graphs

10.37236/9638 ◽  
2021 ◽  
Vol 28 (1) ◽  
Author(s):  
Dan Hu ◽  
Hajo Broersma ◽  
Jiangyou Hou ◽  
Shenggui Zhang
Keyword(s):  
Error Bounds ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Undirected Graph ◽  
Laplacian Matrix ◽  
Probability Inequality ◽  
Mixed Graph ◽  
Mixed Graphs

A mixed graph is a graph that can be obtained from a simple undirected graph by replacing some of the edges by arcs in precisely one of the two possible directions. The Hermitian adjacency matrix of a mixed graph $G$ of order $n$ is the $n \times n$ matrix $H(G)=(h_{ij})$, where $h_{ij}=-h_{ji}= \boldsymbol{\mathrm{i}}$ (with $\boldsymbol{\mathrm{i}} =\sqrt{-1})$ if there exists an arc from $v_i$ to $v_j$ (but no arc from $v_j$ to $v_i$), $h_{ij}=h_{ji}=1$ if there exists an edge (and no arcs) between $v_i$ and $v_j$, and $h_{ij}= 0$ otherwise (if $v_i$ and $v_j$ are neither joined by an edge nor by an arc). We study the spectra of the Hermitian adjacency matrix and the normalized Hermitian Laplacian matrix of general random mixed graphs, i.e., in which all arcs are chosen independently with different probabilities (and an edge is regarded as two oppositely oriented arcs joining the same pair of vertices). For our first main result, we derive a new probability inequality and apply it to obtain an upper bound on the eigenvalues of the Hermitian adjacency matrix. Our second main result shows that the eigenvalues of the normalized Hermitian Laplacian matrix can be approximated by the eigenvalues of a closely related weighted expectation matrix, with error bounds depending on the minimum expected degree of the underlying undirected graph.

scholarly journals Skew-rank of an oriented graph in terms of the rank and dimension of cycle space of its underlying graph

Filomat ◽  
2018 ◽  
Vol 32 (4) ◽  
pp. 1303-1312 ◽  
Author(s):  
Yong Lu ◽  
Ligong Wang ◽  
Qiannan Zhou
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Adjacency Matrix ◽  
Oriented Graph ◽  
Connected Components ◽  
Cycle Space ◽  
Vertex Number ◽  
Underlying Graph ◽  
Skew Adjacency Matrix

Let G? be an oriented graph and S(G?) be its skew-adjacency matrix, where G is called the underlying graph of G?. The skew-rank of G?, denoted by sr(G?), is the rank of S(G?). Denote by d(G) = |E(G)|-|V(G)| + ?(G) the dimension of cycle spaces of G, where |E(G)|, |V(G)| and ?(G) are the edge number, vertex number and the number of connected components of G, respectively. Recently, Wong, Ma and Tian [European J. Combin. 54 (2016) 76-86] proved that sr(G?) ? r(G) + 2d(G) for an oriented graph G?, where r(G) is the rank of the adjacency matrix of G, and characterized the graphs whose skew-rank attain the upper bound. However, the problem of the lower bound of sr(G?) of an oriented graph G? in terms of r(G) and d(G) of its underlying graph G is left open till now. In this paper, we prove that sr(G?) ? r(G)-2d(G) for an oriented graph G? and characterize the graphs whose skew-rank attain the lower bound.

A NOTE ON GENERALIZED RAINBOW CONNECTION OF CONNECTED GRAPHS AND THEIR NUMBER OF EDGES

2021 ◽  
Vol 66 (3) ◽  
pp. 3-7
Author(s):  
Anh Nguyen Thi Thuy ◽  
Duyen Le Thi
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Connection Number ◽  
Connected Graphs ◽  
Rainbow Connection Number ◽  
Rainbow Connection ◽  
Rainbow Path ◽  
Vertex Disjoint

Let l ≥ 1, k ≥ 1 be two integers. Given an edge-coloured connected graph G. A path P in the graph G is called l-rainbow path if each subpath of length at most l + 1 is rainbow. The graph G is called (k, l)-rainbow connected if any two vertices in G are connected by at least k pairwise internally vertex-disjoint l-rainbow paths. The smallest number of colours needed in order to make G (k, l)-rainbow connected is called the (k, l)-rainbow connection number of G and denoted by rck,l(G). In this paper, we first focus to improve the upper bound of the (1, l)-rainbow connection number depending on the size of connected graphs. Using this result, we characterize all connected graphs having the large (1, 2)-rainbow connection number. Moreover, we also determine the (1, l)-rainbow connection number in a connected graph G containing a sequence of cut-edges.

scholarly journals A New Bound on the Domination Number of Graphs with Minimum Degree Two

10.37236/499 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Michael A. Henning ◽  
Ingo Schiermeyer ◽  
Anders Yeo
Keyword(s):  
Connected Graph ◽  
Minimum Degree ◽  
Domination Number ◽  
Degree Condition ◽  
Cut Vertex ◽  
Delta G ◽  
Vertex Disjoint ◽  
Special Cycle

For a graph $G$, let $\gamma(G)$ denote the domination number of $G$ and let $\delta(G)$ denote the minimum degree among the vertices of $G$. A vertex $x$ is called a bad-cut-vertex of $G$ if $G-x$ contains a component, $C_x$, which is an induced $4$-cycle and $x$ is adjacent to at least one but at most three vertices on $C_x$. A cycle $C$ is called a special-cycle if $C$ is a $5$-cycle in $G$ such that if $u$ and $v$ are consecutive vertices on $C$, then at least one of $u$ and $v$ has degree $2$ in $G$. We let ${\rm bc}(G)$ denote the number of bad-cut-vertices in $G$, and ${\rm sc}(G)$ the maximum number of vertex disjoint special-cycles in $G$ that contain no bad-cut-vertices. We say that a graph is $(C_4,C_5)$-free if it has no induced $4$-cycle or $5$-cycle. Bruce Reed [Paths, stars and the number three. Combin. Probab. Comput. 5 (1996), 277–295] showed that if $G$ is a graph of order $n$ with $\delta(G) \ge 3$, then $\gamma(G) \le 3n/8$. In this paper, we relax the minimum degree condition from three to two. Let $G$ be a connected graph of order $n \ge 14$ with $\delta(G) \ge 2$. As an application of Reed's result, we show that $\gamma(G) \le \frac{1}{8} ( 3n + {\rm sc}(G) + {\rm bc}(G))$. As a consequence of this result, we have that (i) $\gamma(G) \le 2n/5$; (ii) if $G$ contains no special-cycle and no bad-cut-vertex, then $\gamma(G) \le 3n/8$; (iii) if $G$ is $(C_4,C_5)$-free, then $\gamma(G) \le 3n/8$; (iv) if $G$ is $2$-connected and $d_G(u) + d_G(v) \ge 5$ for every two adjacent vertices $u$ and $v$, then $\gamma(G) \le 3n/8$. All bounds are sharp.

scholarly journals Partitioning the Power Set of $[n]$ into $C_k$-free parts

10.37236/8385 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Eben Blaisdell ◽  
András Gyárfás ◽  
Robert A. Krueger ◽  
Ronen Wdowinski
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Type Problem ◽  
Color Class ◽  
Power Set ◽  
Ramsey Type ◽  
Partition Class ◽  
Cycle Path

We show that for $n \geq 3, n\ne 5$, in any partition of $\mathcal{P}(n)$, the set of all subsets of $[n]=\{1,2,\dots,n\}$, into $2^{n-2}-1$ parts, some part must contain a triangle — three different subsets $A,B,C\subseteq [n]$ such that $A\cap B,A\cap C,B\cap C$ have distinct representatives. This is sharp, since by placing two complementary pairs of sets into each partition class, we have a partition into $2^{n-2}$ triangle-free parts.  We also address a more general Ramsey-type problem: for a given graph $G$, find (estimate) $f(n,G)$, the smallest number of colors needed for a coloring of $\mathcal{P}(n)$, such that no color class contains a Berge-$G$ subhypergraph. We give an upper bound for $f(n,G)$ for any connected graph $G$ which is asymptotically sharp when $G$ is a cycle, path, or star. Additional bounds are given when $G$ is a $4$-cycle and when $G$ is a claw.

On the relation between theH-rank of a mixed graph and the matching number of its underlying graph

2017 ◽  
Vol 66 (9) ◽  
pp. 1853-1869 ◽  
Author(s):  
Chen Chen ◽  
Jing Huang ◽  
Shuchao Li
Keyword(s):  
Matching Number ◽  
Mixed Graph ◽  
Underlying Graph

The relation between the H-rank of a mixed graph and the independence number of its underlying graph

2018 ◽  
Vol 67 (11) ◽  
pp. 2230-2245 ◽  
Author(s):  
Shuchao Li ◽  
Siqi Zhang ◽  
Baogen Xu
Keyword(s):  
Independence Number ◽  
Mixed Graph ◽  
Underlying Graph

scholarly journals The Unique Mixed Almost Moore Graph with Parameters k = 2, r = 2 and z = 1

2017 ◽  
Vol 17 (03n04) ◽  
pp. 1741005 ◽  
Author(s):  
DOMINIQUE BUSET ◽  
NACHO LÓPEZ ◽  
JOSEP M. MIRET
Keyword(s):  
Upper Bound ◽  
Mixed Graph ◽  
Moore Graph

A natural upper bound for the maximum number of vertices in a mixed graph with maximum undirected degree r, maximum directed out-degree z and diameter k is given by the mixed Moore bound. Graphs with order attaining the Moore bound are known as Moore graphs, and they are very rare. Besides, graphs with prescribed parameters and order one less than the corresponding Moore bound are known as almost Moore graphs. In this paper we prove that there is a unique mixed almost Moore graph of diameter k = 2 and parameters r = 2 and z = 1.

scholarly journals On Total Regularity of Mixed Graphs with Order Close to the Moore Bound

2019 ◽  
Vol 35 (6) ◽  
pp. 1253-1272
Author(s):  
James Tuite ◽  
Grahame Erskine
Keyword(s):  
Open Problem ◽  
Mixed Graph ◽  
Difficult Question ◽  
Moore Graph ◽  
Network Topologies ◽  
Mixed Graphs

Abstract The undirected degree/diameter and degree/girth problems and their directed analogues have been studied for many decades in the search for efficient network topologies. Recently such questions have received much attention in the setting of mixed graphs, i.e. networks that admit both undirected edges and directed arcs. The degree/diameter problem for mixed graphs asks for the largest possible order of a network with diameter k, maximum undirected degree $$\le r$$≤r and maximum directed out-degree $$\le z$$≤z. Similarly one can search for the smallest possible k-geodetic mixed graphs with minimum undirected degree $$\ge r$$≥r and minimum directed out-degree $$\ge z$$≥z. A simple counting argument reveals the existence of a natural bound, the Moore bound, on the order of such graphs; a graph that meets this limit is a mixed Moore graph. Mixed Moore graphs can exist only for $$k = 2$$k=2 and even in this case it is known that they are extremely rare. It is therefore of interest to search for graphs with order one away from the Moore bound. Such graphs must be out-regular; a much more difficult question is whether they must be totally regular. For $$k = 2$$k=2, we answer this question in the affirmative, thereby resolving an open problem stated in a recent paper of López and Miret. We also present partial results for larger k. We finally put these results to practical use by proving the uniqueness of a 2-geodetic mixed graph with order exceeding the Moore bound by one.

A Note on the Diameter of a Graph

1968 ◽  
Vol 11 (3) ◽  
pp. 499-501 ◽  
Author(s):  
J. A. Bondy
Keyword(s):  
Shortest Path ◽  
Upper Bound ◽  
Connected Graph ◽  
Maximum Distance ◽  
Image Position

The distance d(x, y) between vertices x, y of a graph G is the length of the shortest path from x to y in G. The diameter δ(G) of G is the maximum distance between any pair of vertices in G. i.e. δ(G) = max max d(x, y). In this note we obtain an upper boundx ε G y ε Gfor δ(G) in terms of the numbers of vertices and edges in G. Using this bound it is then shown that for any complement-connected graph G with N verticeswhere is the complement of G.

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