A stable set in a graph G is a set of mutually non-adjacent vertices, alpha(G) is the size of a maximum stable set of G, and core(G) is the intersection of all its maximum stable sets. In this paper we demonstrate that in a tree T, of order n greater than 1, any stable set of size greater or equal to n/2 contains at least one pendant vertex. Hence, we deduce that any maximum stable set in a tree contains at least one pendant vertex. Our main finding is the theorem claiming that if T does not own a perfect matching, then at least two pendant vertices an even distance apart belong to core(T). While it is proved by Levit and Mandrescu that if G is a connected bipartite graph of order at least 2, then the size of core(G) is different from 1, our new statement reveals an additional structure of the intersection of all maximum stable sets of a tree. The above assertions give refining of one result of Hammer, Hansen and Simeone, stating that if a graph G is of order less than 2*alpha(G), then core(G) is non-empty, and also of a result of Jamison, Gunter, Hartnel and Rall, and Zito, saying that for a tree T of order at least two, the size of core(G) is different from 1.
In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.
The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.
The Sombor index is a topological index in graph theory defined by Gutman in 2021. In this article we find the maximum Sombor index of unicyclic graphs with a fixed number of pendant vertices. We also provide the unique graph among the chosen class where the maximum Sombor index is attained.
In this paper we determine the graph whose least eigenvalue of signless Laplacian attains the minimum or maximum among all connected non-bipartite graphs of fixed order and given number of pendant vertices. Thus we obtain a lower bound and an upper bound for the least eigenvalue of signless Laplacian of a graph in terms of the number of pendent vertices.
For a graph $G$, the first multiplicative Zagreb index $\prod_1(G) $ is the product of squares of vertex degrees, and the second multiplicative Zagreb index $\prod_2(G) $ is the product of products of degrees of pairs of adjacent vertices. In this paper, we explore graphs with extremal $Π_{1}(G)$ and $Π_{2}(G)$ in terms of (edge) connectivity and pendant vertices. The corresponding extremal graphs are characterized with given connectivity at most $k$ and $p$ pendant vertices. In addition, the maximum and minimum values of $\prod_1(G) $ and $\prod_2(G) $ are provided. Our results extend and enr…
One theorem of Nemhauser and Trotter ensures that, under certain conditions, a stable set of a graph G can be enlarged to a maximum stable set of this graph. For example, any stable set consisting of only simplicial vertices is contained in a maximum stable set of G. In this paper we demonstrate that an inverse assertion is true for trees of order greater than one, where, in fact, all the simplicial vertices are pendant. Namely, we show that any maximum stable set of such a tree contains at least one pendant vertex. Moreover, we prove that if T does not own a perfect matching, then a stable se…
The total eccentricity index of a connected graph is defined as sum of the eccentricities of all its vertices. We denote the set of all connected graphs on $n$ vertices with $k$ pendant vertices by $\mathfrak{H}_{n,k}$ and denote the set of all connected graphs on $n$ vertices with $s$ cut vertices by $\mathfrak{C_{n,s}}$. In this paper, we give the sharp lower and upper bounds on the total eccentricity index over $\mathfrak{H}_{n,k}$ and the sharp lower bound for the same over $\mathfrak{C_{n,s}}$. We also provide the sharp upper bounds on the total eccentricity index over $\mathfrak{C_{n,s}}…
The Steiner tree problem in graphs has applications in network design or circuit layout. Given a set $S$ of vertices, $|S| \geq 2,$ a tree connecting all vertices of $S$ is called an $S$-Steiner tree (tree connecting $S$). The reliability of a network $G$ to connect any $S$ vertices ($|S|$ number of vertices) in $G$ can be measure by this parameter. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then that tree is called a pendant S-Steiner tree. Two pendant $S$-Steiner trees $T$ and $T'$ are said to be internally disjoint if $E(T) \cap E(T') = \emptyset$ and $V(T…
A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the correspon…
A tree is called double starlike if it has exactly two vertices of degree greater than two. Let $H(p,n,q)$ denote the double starlike tree obtained by attaching $p$ pendant vertices to one pendant vertex of the path $P_n$ and $q$ pendant vertices to the other pendant vertex of $P_n$. In this paper, we prove that $H(p,n,q)$ is determined by its Laplacian spectrum.
An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. Let $χ(G)$ be the chromatic number of $G$. In this paper, sharp upper and lower bounds of $χ_{la}(G)$ for $G$ with pend…
Let $G$ be a simple graph. A pendant path of $G$ is a path such that one of its end vertices has degree $1$, the other end has degree $\ge3$, and all the internal vertices have degree $2$. Let $p_k(G)$ be the number of pendant paths of length $k$ of $G$, and $q_k(G)$ be the number of vertices with degree $\ge3$ which are an end vertex of some pendant paths of length $k$. Motivated by the problem of characterizing dendritic trees, N. Saito and E. Woei conjectured that any graph $G$ has some Laplacian eigenvalue with multiplicity at least $p_k(G)-q_k(G)$. We prove a more general result for both …
For some positive integer $k$, if the finite cyclic group $\mathbb{Z}_k$ can act freely on a graph $G$, then we say that $G$ is $k$-symmetric. In 1985, Faria showed that the multiplicity of Laplacian eigenvalue 1 is greater than or equal to the difference between the number of pendant vertices and the number of quasi-pendant vertices. But if a graph has a pendant vertex, then it is at most 1-connected. In this paper, we investigate a class of 2-connected $k$-symmetric graphs with a Laplacian eigenvalue 1. We also identify a class of $k$-symmetric graphs in which all Laplacian eigenvalues are i…
The Kemeny's constant $κ(G)$ of a connected undirected graph $G$ can be interpreted as the expected transit time between two randomly chosen vertices for the Markov chain associated with $G$. In certain cases, inserting a new edge into $G$ has the counter-intuitive effect of increasing the value of $κ(G)$. In the current work we identify a large class of graphs exhibiting this "paradoxical" behavior - namely, those graphs having a pair of twin pendant vertices. We also investigate the occurrence of this phenomenon in random graphs, showing that almost all connected planar graphs are paradoxica…
For a tree $T$, let $i_T(t)$ be the number of independent sets of size $t$ in $T$. It is an open question, raised by Alavi, Malde, Schwenk and Erdős, whether the sequence $(i_T(t))_{t \geq 0}$ is always unimodal. Here we answer the question in the affirmative for some recursively defined families of trees, specifically paths with auxiliary trees dropped from the vertices in a periodic manner. In particular, extending a result of Wang and B.-X. Zhu, we show unimodality of the independent set sequence of a path on $2n$ vertices with $\ell_1$ and $\ell_2$ pendant edges dropped alternately from th…
For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we call $F(G),$ the core index of $G$. In this paper, we characterize the graphs which extremize the core index among all graphs on $n$ vertices with $k\geq 0$ connected components. We extend our study of core index to unicyclic graphs and connected graphs with fixed number of pendant vertices. We obtained the unicyclic graphs which extremize the core index over …
The concept of pedant tree-connectivity was introduced by Hager in 1985. For a graph $G=(V,E)$ and a set $S\subseteq V(G)$ of at least two vertices, \emph{an $S$-Steiner tree} or \emph{a Steiner tree connecting $S$} (or simply, \emph{an $S$-tree}) is a such subgraph $T=(V',E')$ of $G$ that is a tree with $S\subseteq V'$. For an $S$-Steiner tree, if the degree of each vertex in $S$ is equal to one, then this tree is called a \emph{pedant $S$-Steiner tree}. Two pedant $S$-Steiner trees $T$ and $T'$ are said to be \emph{internally disjoint} if $E(T)\cap E(T')=\varnothing$ and $V(T)\cap V(T')=S$. …
We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-…
In this paper we prove that for a general tree $T$, if $A$ is T-TP, all the submatrices of $A$ associated with the deletion of pendant vertices are $P$-matrices, and $\det A>0$, then the smallest eigenvalue has an eigenvector signed according to $T$.
A family of sets is intersecting if every pair of its sets intersect. A star is a family with some element (a center) in each of its sets. The classical 1961 result of Erdős, Ko, and Rado states that every intersecting family of r-sets with $r\leq n/2$ has size at most that of a star. We say that graph G is r-EKR if, among all intersecting families of independent r-sets of G, the largest is attained by a star. In 2005 Holroyd and Talbot conjectured that every graph G is r-EKR for all $1\leq r\leq μ(G)/2$, where $μ(G)$ is the size of the smallest maximal independent set in G. We verified the co…
For $n\geq 5$ and $2\leq g\leq n-3,$ consider the tree $P_{n-g,g}$ on $n$ vertices which is obtained by adding $g$ pendant vertices to one degree $1$ vertex of the path $P_{n-g}$. We call the trees $P_{n-g,g}$ as path-star trees. We prove that over all trees on $n\geq 5$ vertices, the distance between center and subtree core and the distance between centroid and subtree core are maximized by some path-star trees. We also prove that the tree $P_{n-g_0,g_0}$ maximizes both the distances among all path-star trees on $n$ vertices, where $g_0$ is the smallest positive integer such that $2^{g_0}+g_0…
In this paper, new graphs $G_τ=\left(V,E\right)$ are constructed from the discrete topological space $(X,τ)\ $ . Several properties of this type of graphs are given such that: the clique number equals the number of elements in X also the number of pendants vertices, $G_τ$ has no isolated vertices, the minimum degree in $G_τ$ is one and maximum degree equal $n-1+\sum^{n-1}_{i=2}\binom{n-1}{i}$ , the minimum dominating set is determined and $γ(G_τ)$ is evaluated for $G_τ$ and for corona and join operations between to discrete topological graphs. At what matter $β\left(G_τ\right)=γ(G_τ)$ is discu…
Denote by $L_{g, l}$ the $lollipop$ $graph$ obtained by attaching a pendant path $\mathbb{P}=v_{g}v_{g+1}\cdots v_{g+l}$ ($l\geq 1$) to a cycle $\mathbb{C}=v_{1}v_{2}\cdots v_{g}v_{1}$ ($g\geq 3$). A $\mathcal {F}_{g, l}$-$graph$ of order $n\geq g+1$ is defined to be the graph obtained by attaching $n-g-l$ pendent vertices to some of the nonpendant vertices of $L_{g, l}$ in which each vertex other than $v_{g+l-1}$ is attached at most one pendant vertex. A $\mathcal {F}^{\circ}_{g, l}$-graph is a $\mathcal {F}_{g, l}$-$graph$ in which $v_{g}$ is attached with pendant vertex. Denote by $q_{min}$…
For a set of non-negative integers $L$, the $L$-intersection number of a graph is the smallest number $l$ for which there is an assignment on the vertices to subsets $A_v \subseteq \{1,\dots, l\}$, such that every two vertices $u,v$ are adjacent if and only if $|A_u \cap A_v|\in L$. The bipartite $L$-intersection number is defined similarly when the conditions are considered only for the vertices in different parts. In this paper, some lower bounds for the (bipartite) $L$-intersection number of a graph for various types $L$ in terms of the minimum rank of graph are obtained.
We show that every pair of longest paths in a $k$-connected graph on $n$ vertices intersect each other in at least $(8k-n+2)/5$ vertices. We also show that, in a 4-connected graph, every pair of longest paths intersect each other in at least four vertices. This confirms a conjecture of Hippchen for $k$-connected graphs when $k\leq 4$ or $k\geq (n-2)/3$.
For a vector space $V$ the \emph{intersection graph of subspaces} of $V$, denoted by $G(V)$, is the graph whose vertices are in a one-to-one correspondence with proper nontrivial subspaces of $V$ and two distinct vertices are adjacent if and only if the corresponding subspaces of $V$ have a nontrivial (nonzero) intersection. In this paper, we study the clique number, the chromatic number, the domination number and the independence number of the intersection graphs of subspaces of a vector space.
For each $n\in\mathbb{N}$, let $\mathbf{Q}_n$ be a uniform rooted measured quadrangulation of size $n$ conditioned to have $r(n)$ vertices in its root block. We prove that for a suitable function $r(n)$, after rescaling graph distance by $\left(\frac{21}{40\cdot r(n)}\right)^{1/4}$, with an appropriate rescaling of measure, $\mathbf{Q}_n$ converges to a random pointed measured non-compact metric space $\boldsymbol{ \mathcal{S}}$, in the local Gromov-Hausdorff-Prokhorov topology; the space $\boldsymbol{ \mathcal{S}}$ is built by identifying a uniform point of the Brownian map with the distingui…
Random intersection graphs are characterized by three parameters: $n$, $m$ and $p$, where $n$ is the number of vertices, $m$ is the number of objects, and $p$ is the probability that a given object is associated with a given vertex. Two vertices in a random intersection graph are adjacent if and only if they have an associated object in common. When $m=\lfloor n^α\rfloor$ for constant $α$, we provide a condition, called {\em strictly $α$-balanced}, for the Poisson convergence of the number of induced copies of a fixed subgraph.
The Second Neighborhood Conjecture states that every simple digraph has a vertex whose second out-neighborhood is at least as large as its first out-neighborhood, i.e. a vertex with the Second Neighborhood Property. A cycle intersection graph of an even graph is a new graph whose vertices are the cycles in a cycle decomposition of the original graph and whose edges represent vertex intersections of the cycles. By using a digraph variant of this concept, we prove that Eulerian digraphs which admit a simple cycle intersection graph have not only adhere to the Second Neighborhood Conjecture, but …
Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes of which $c$ are cut vertices. For all valid $n,c$, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of $n,c$, however, there is a unique maximal unicyclic graph with $n$ …
Let claw be the graph $K_{1,3}$. A graph $G$ on $n\geq 3$ vertices is called \emph{o}-heavy if each induced claw of $G$ has a pair of end-vertices with degree sum at least $n$, and 1-heavy if at least one end-vertex of each induced claw of $G$ has degree at least $n/2$. In this note, we show that every 2-connected $o$-heavy or 3-connected 1-heavy graph is Hamiltonian if we restrict Fan-type degree condition or neighborhood intersection condition to certain pairs of vertices in some small induced subgraphs of the graph. Our results improve or extend previous results of Broersma et al., Chen et …
In this paper we will show that there does not exist a distance-regular graph $Γ$ with intersection array $\{80, 54,12; 1, 6, 60\}$. We first show that a local graph $Δ$ of $Γ$ does not contain a coclique with 5 vertices, and then we prove that the graph $Γ$ is geometric by showing that $Δ$ consists of 4 disjoint cliques with 20 vertices. Then we apply a result of Koolen and Bang to the graph $Γ$, and we could obtain that there is no such a distance-regular graph.
Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these intersections are different for different vertices $x$. Let $k$ be a positive integer. We will consider graphs where \emph{every} $k$-subset is identifying. We prove that for every $k>1$ the maximal order of such a graph is at most $2k-2.$ Constructions attaining the maximal order are given for infinitely many values of $k.$ The corresponding problem of $k$-subs…
Consider a directed graph (digraph) in which vertices are assigned color sets, and two vertices are connected if and only if they share at least one color and the tail vertex has a strictly smaller color set than the head. We seek to determine the smallest possible size of the union of the color sets that allows for such a digraph representation. To address this problem, we introduce the new notion of a directed intersection representation of a digraph, and show that it is well-defined for all directed acyclic graphs (DAGs). We then proceed to introduce the directed intersection number (DIN), …
A graph $G$ has $p$-intersection number at most $d$ if it is possible to assign to every vertex $u$ of $G$, a subset $S(u)$ of some ground set $U$ with $|U|=d$ in such a way that distinct vertices $u$ and $v$ of $G$ are adjacent in $G$ if and only if $|S(u)\cap S(v)|\geq p$. We show that every minimal forbidden induced subgraph for the hereditary class ${\cal G}(d,p)$ of graphs whose $p$-intersection number is at most $d$, has order at most $3\cdot 2^{d+1}+1$, and that the exponential dependence on $d$ in this upper bound is necessary. For $p\in \{ d-1,d-2\}$, we provide more explicit results …
For a connected graph $G:=(V,E)$, the Steiner distance $d_G(X)$ among a set of vertices $X$ is the minimum size among all the connected subgraphs of $G$ whose vertex set contains $X$. The $k-$Steiner distance matrix $D_k(G)$ of $G$ is a matrix whose rows and columns are indexed by $k-$subsets of $V$. For $k$-subsets $X_1$ and $X_2$, the $(X_1,X_2)-$entry of $D_k(G)$ is $d_G(X_1 \cup X_2)$. In this paper, we show that the rank of $2-$Steiner distance matrix of a caterpillar graph on $N$ vertices and with $p$ pendant veritices is $2N-p-1$.
It was noted already in the 90s that many classic graph classes, such as interval, chordal, and bipartite graphs, can be characterized by the existence of an ordering of the vertices avoiding some ordered subgraphs, called patterns. Very recently, all the classes corresponding to patterns on three vertices (including the ones mentioned above) have been listed, and proved to be efficiently recognizable. In contrast, very little is known about patterns on four vertices. One of the few graph classes characterized by a pattern on four vertices is the class of intersection graphs of rectangles that…
We prove that for a connected simple graph $G$ with $n\le 10$ vertices, and two longest paths $C$ and $D$ in $G$, the intersection of vertex sets $V(C)\cap V(D)$ is a separator. This shows that the graph found previously with $n=11$, in which the complement of the intersection of vertex sets $V(C)\cap V(D)$ of two longest paths is connected, is minimal.
A semiregular tree is a tree where all non-pendant vertices have the same degree. Belardo et al. (MATCH Commun. Math. Chem. 61(2), pp. 503-515, 2009) have shown that among all semiregular trees with a fixed order and degree, a graph with index is a caterpillar. In this technical report we provide a different proof for this theorem. Furthermore, we give counter examples that show this result cannot be generalized to the class of trees with a given (non-constant) degree sequence.
We prove tight upper bounds for the number of vertices of a simple polygon that is the union or the intersection of two simple polygons with given numbers of convex and concave vertices. The similar question on graphs of the lower (or upper) envelope of two continuous piecewise linear functions is considered.
We consider the typical distance between vertices of the giant component of a random intersection graph having a power law (asymptotic) vertex degree distribution with infinite second moment. Given two vertices from the giant component we construct O(log log n) upper bound (in probability) for the length of the shortest path connecting them.
We show that every graph of maximum degree 3 can be represented as the intersection graph of axis parallel boxes in three dimensions, that is, every vertex can be mapped to an axis parallel box such that two boxes intersect if and only if their corresponding vertices are adjacent. In fact, we construct a representation in which any two intersecting boxes just touch at their boundaries. Further, this construction can be realized in linear time.
