Saieed Akbari

Let $G$ be a graph of order $n$ with eigenvalues $\lambda_1 \geq \cdots \geq\lambda_n$. Let \[s^+(G)=\sum_{\lambda_i>0} \lambda_i^2, \qquad s^-(G)=\sum_{\lambda_i<0} \lambda_i^2.\] The smaller value, $s(G)=\min\{s^+(G), s^-(G)\}$ is called the square energy of $G$. In 2016, Elphick, Farber, Goldberg, and Wocjan conjectured that for every connected graph $G$ of order $n$, $s(G)\geq n-1.$ No linear bound for $s(G)$ in terms of $n$ is known. Let $H_1, \ldots, H_k$ be disjoint induced subgraphs of $G$. In this note, we prove that \[s^+(G)\geq\sum_{i=1}^{k} s^+(H_i) \quad \text{ and } \quad s^-(G)\geq\sum_{i=1}^{k} s^-(H_i),\] and then use this result to prove that $s(G)\geq \frac{3n}{4}$ for every connected graph $G$ of order $n\ge 4$.

Let $G=(V,E)$ be a simple graph. A dominating set of $G$ is a subset $S\subseteq V$ such that every vertex not in $S$ is adjacent to at least one vertex in $S$. The cardinality of a smallest dominating set of $G$, denoted by $\gamma(G)$, is the domination number of $G$. Two vertices are neighbors if they are adjacent. A super dominating set is a dominating set $S$ with the additional property that every vertex in $V \setminus S$ has a neighbor in $S$ that is adjacent to no other vertex in $V \setminus S$. Moreover if every vertex in $V \setminus S$ has degree at least~$2$, then $S$ is an end super dominating set. The end super domination number is the minimum cardinality of an end super dominating set. We give applications of end super dominating sets as main servers and temporary servers of networks. We determine the exact value of the end super domination number for specific classes of graphs, and we count the number of end super dominating sets in these graphs. Tight upper bounds on the end super domination number are established, where the graph is modified by vertex (edge) removal and contraction.

The chromatic vertex (resp.\ edge) stability number ${\rm vs}_{\chi}(G)$ (resp.\ ${\rm es}_{\chi}(G)$) of a graph $G$ is the minimum number of vertices (resp.\ edges) whose deletion results in a graph $H$ with $\chi(H)=\chi(G)-1$. In the main result it is proved that if $G$ is a graph with $\chi(G) \in \{ \Delta(G), \Delta(G)+1 \}$, then ${\rm vs}_{\chi}(G) = {\rm ivs}_{\chi}(G)$, where ${\rm ivs}_{\chi}(G)$ is the independent chromatic vertex stability number. The result need not hold for graphs $G$ with $\chi(G) \le \frac{\Delta(G)+1}{2}$. It is proved that if $\chi(G) > \frac{\Delta(G)}{2}+1$, then ${\rm vs}_{\chi}(G) = {\rm es}_{\chi}(G)$. A Nordhaus-Gaddum-type result on the chromatic vertex stability number is also given.

An induced path factor of a graph $G$ is a set of induced paths in $G$ with the property that every vertex of $G$ is in exactly one of the paths. The induced path number $\rho(G)$ of $G$ is the minimum number of paths in an induced path factor of $G$. We show that if $G$ is a connected cubic graph on $n>6$ vertices, then $\rho(G)\le(n-1)/3$. Fix an integer $k\ge3$. For each $n$, define $\mathcal{M}_n$ to be the maximum value of $\rho(G)$ over all connected $k$-regular graphs $G$ on $n$ vertices. As $n\rightarrow\infty$ with $nk$ even, we show that $c_k=\lim(\mathcal{M}_n/n)$ exists. We prove that $5/18\le c_3\le1/3$ and $3/7\le c_4\le1/2$ and that $c_k=\frac12-O(k^{-1})$ for $k\rightarrow\infty$.

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$. If, in addition, $S$ is an independent set, then $S$ is an independent dominating set. The independent domination number $i(G)$ of $G$ is the minimum cardinality of an independent dominating set in $G$. In 2013 Goddard and Henning [Discrete Math 313 (2013), 839--854] conjectured that if $G$ is a connected cubic graph of order $n$, then $i(G) \le \frac{3}{8}n$, except if $G$ is the complete bipartite graph $K_{3,3}$ or the $5$-prism $C_5 \, \Box \, K_2$. Further they construct two infinite families of connected cubic graphs with independent domination three-eighths their order. They remark that perhaps it is even true that for $n > 10$ these two families are only families for which equality holds. In this paper, we provide a new family of connected cubic graphs $G$ of order $n$ such that $i(G) = \frac{3}{8}n$. We also show that if $G$ is a subcubic graph of order $n$ with no isolated vertex, then $i(G) \le \frac{1}{2}n$, and we characterize the graphs achieving equality in this bound.

Let $G$ be a graph, a dominating induced matching (DIM) of $G$ is an induced matching that dominates every edge of $G$. In this paper we show that if a graph $G$ has a DIM, then $\chi(G) \leqslant 3$. Also, it is shown that if $G$ is a connected graph whose all edges can be partitioned into DIM, then $G$ is either a regular graph or a biregular graph and indeed we characterize all graphs whose edge set can be partitioned into DIM. Also, we prove that if $G$ is an $r$-regular graph of order $n$ whose edges can be partitioned into DIM, then $n$ is divisible by $\binom{2r - 1}{r - 1}$ and $n = \binom{2r - 1}{r - 1}$ if and only if $G$ is the Kneser graph with parameters $r-1$, $2r-1$.

Let $G$ be a graph of order $n$. It is well-known that $\alpha(G)\geq \sum_{i=1}^n \frac{1}{1+d_i}$, where $\alpha(G)$ is the independence number of $G$ and $d_1,\ldots,d_n$ is the degree sequence of $G$. We extend this result to digraphs by showing that if $D$ is a digraph with $n$ vertices, then $ \alpha(D)\geq \sum_{i=1}^n \left( \frac{1}{1+d_i^+} + \frac{1}{1+d_i^-} - \frac{1}{1+d_i}\right)$, where $\alpha(D)$ is the maximum size of an acyclic vertex set of $D$. Golowich proved that for any digraph $D$, $\chi(D)\leq \lceil \frac{4k}{5} \rceil+2$, where $k=max(\Delta^+(D),\Delta^-(D))$. We give a short and simple proof for this result. Next, we investigate the chromatic number of tournaments and determine the unique tournament such that for every integer $k>1$, the number of proper $k$-colorings of that tournament is maximum among all strongly connected tournaments with the same number of vertices. Also, we find the chromatic polynomial of the strongly connected tournament with the minimum number of cycles.

Let $G$ be a group‎. ‎The power graph of $G$ is a graph with the vertex‎ ‎set $G$‎, ‎having an edge between two elements whenever one is a power of the other‎. ‎We characterize nilpotent groups whose power graphs have finite independence number‎. ‎For a bounded exponent group‎, ‎we prove its power graph is a perfect graph and we determine‎ ‎its clique/chromatic number‎. ‎Furthermore‎, ‎it is proved that for every group $G$‎, ‎the clique number of the power graph of $G$ is at most countably infinite‎. ‎We also measure how close the power graph is to the commuting graph by introducing a new graph which lies in between‎. ‎We call this new graph as the enhanced power graph‎. ‎For an arbitrary pair of these three graphs we characterize finite groups for which this pair of graphs are equal‎.

Let $G$ be a graph. Assume that $l$ and $k$ are two natural numbers. An $l$-sum flow on a graph $G$ is an assignment of non-zero real numbers to the edges of $G$ such that for every vertex $v$ of $G$ the sum of values of all edges incidence with $v$ equals $l$. An $l$-sum $k$-flow is an $l$-sum flow with values from the set $\{\pm 1,\ldots ,\pm(k-1)\}$. Recently, it was proved that for every $r, r\geq 3$, $r\neq 5$, every $r$-regular graph admits a $0$-sum $5$-flow. In this paper we settle a conjecture by showing that every $5$-regular graph admits a $0$-sum $5$-flow. Moreover, we prove that every $r$-regular graph of even order admits a $1$-sum $5$-flow.