arXiv subjects

Suil O

Publications and source records attributed to Suil O.

24 records · Page 2Linked to original sources

arXiv2017

Sharp bounds for the Randic index of graphs with given minimum and maximum degree

The Randi{\' c} index of a graph $G$, written $R(G)$, is the sum of $\frac 1{\sqrt{d(u)d(v)}}$ over all edges $uv$ in $E(G)$. %let $R(G)=\sum_{uv \in E(G)} \frac 1{\sqrt{d(u)d(v)}}$, which is called the Randi{\' c} index of it. Let $d$ and $D$ be positive integers $d < D$. In this paper, we prove that if $G$ is a graph with minimum degree $d$ and maximum degree $D$, then $R(G) \ge \frac{\sqrt{dD}}{d+D}n$; equality holds only when $G$ is an $n$-vertex $(d,D)$-biregular. Furthermore, we show that if $G$ is an $n$-vertex connected graph with minimum degree $d$ and maximum degree $D$, then $R(G) \le \frac n2- \sum_{i=d}^{D-1}\frac 12 \left( \frac 1{\sqrt{i}} - \frac 1{\sqrt{i+1}}\right)^2$; it is sharp for infinitely many $n$, and we characterize when equality holds in the bound.

math.CO↗

arXiv2016

The second largest eigenvalue and vertex-connectivity of regular multigraphs

Let $μ_2(G)$ be the second smallest Laplacian eigenvalue of a graph $G$. The vertex connectivity of $G$, written $κ(G)$, is the minimum size of a vertex set $S$ such that $G-S$ is disconnected. Fiedler proved that $μ_2(G) \le κ(G)$ for a non-complete simple graph $G$; for this reason $μ_2(G)$ is called the "algebraic connectivity" of $G$. We extend his result to multigraphs. For a pair of vertices $u$ and $v$, let $m(u,v)$ be the number of edges with endpoints $u$ and $v$. For a vertex $v$, let $m(v)=\max_{u \in N(v)} m(v,u)$, where $N(v)$ is the set of neighbors of $v$, and let $m(G)=\max_{v \in V(G)} m(v)$. We prove that for any multigraph $G$ whose underlying graph is not a complete graph, $μ_2(G) \le κ(G) m(G)$. We also prove that for any $d$-regular multigraph $G$ whose underlying graph is not the complete graph with 2 vertices, if $μ_2(G) > \frac d4$, then $G$ is 2-connected. For $t\ge2$ and infinitely many $d$, we construct $d$-regular multigraphs $H$ with $μ_2(H)=d$, $κ(H)=t$, and $m(H)=\frac dt$. These graphs show that the inequality $μ_2(G) \le κ(G) m(G)$ is sharp. In addition, we prove that if $G$ is a $d$-regular multigraph whose underlying graph is not a complete graph, then $μ_2(G) \le d$; equality holds for the graphs in the construction.

math.CO↗

arXiv2016

Spectral radius and fractional matchings in graphs

A {\it fractional matching} of a graph $G$ is a function $f$ giving each edge a number in $[0,1]$ so that $\sum_{e \in Γ(v)} f(e) \le 1$ for each $v\in V(G)$, where $Γ(v)$ is the set of edges incident to $v$. The {\it fractional matching number} of $G$, written $α'_*(G)$, is the maximum of $\sum_{e \in E(G)} f(e)$ over all fractional matchings $f$. Let $G$ be an $n$-vertex connected graph with minimum degree $d$, let $λ_1(G)$ be the largest eigenvalue of $G$, and let $k$ be a positive integer less than $n$. In this paper, we prove that if $λ_1(G) < d\sqrt{1+\frac{2k}{n-k}}$, then $α'_*(G) > \frac{n-k}{2}$. As a result, we prove $α'_*(G) \ge \frac{nd^2}{λ_1(G)^2 + d^2}$, we characterize when equality holds in the bound.

math.CO↗

arXiv2015

The difference and ratio of the fractional matching number and the matching number of graphs

Given a graph $G$, the matching number of $G$, written $α'(G)$, is the maximum size of a matching in $G$, and the fractional matching number of $G$, written $α'_f(G)$, is the maximum size of a fractional matching of $G$. In this paper, we prove that if $G$ is an $n$-vertex connected graph that is neither $K_1$ nor $K_3$, then $α'_f(G)-α'(G) \le \frac{n-2}6$ and $\frac{α'_f(G)}{α'(G)} \le \frac{3n}{2n+2}$. Both inequalities are sharp, and we characterize the infinite family of graphs where equalities hold.

math.CO↗

arXiv2014

Edge-connectivity in regular multigraphs from eigenvalues

Let $G$ be a $d$-regular multigraph, and let $λ_2(G)$ be the second largest eigenvalue of $G$. In this paper, we prove that if $λ_2(G) < \frac{d-1+\sqrt{9d^2-10d+17}}4$, then $G$ is 2-edge-connected. Furthermore, for $t\ge2$ we show that $G$ is $(t+1)$-edge-connected when $λ_2(G)<d-t$, and in fact when $λ_2(G)<d-t+1$ if $t$ is odd.

math.CO↗

arXiv2012

Hamiltonicity in connected regular graphs

In 1980, Jackson proved that every 2-connected $k$-regular graph with at most $3k$ vertices is Hamiltonian. This result has been extended in several papers. In this note, we determine the minimum number of vertices in a connected $k$-regular graph that is not Hamiltonian, and we also solve the analogous problem for Hamiltonian paths. Further, we characterize the smallest connected $k$-regular graphs without a Hamiltonian cycle.

math.CO↗