Let $G$ be a graph with order $p$, size $q$ and component number $\omega$. For each $i$ between $p-\omega$ and $q$, let ${𝒞}_i\left(G\right)$ be the family of spanning $i$-edge subgraphs of $G$ with exactly $\omega$ components. For an integer-valued graphical invariant $\varphi$, if $H\to H^{\text{'}}$ is an adjacent edge transformation (AET) implies $|\varphi \left(H\right)-\varphi \left(H^{\text{'}}\right)|\le 1$, then $\varphi$ is said to be continuous with respect to AET. Similarly define the continuity of $\varphi$ with respect to simple edge transformation (SET). Let $M_j\left(\varphi \right)$ and $m_j\left(\varphi \right)$ be the invariants defined by $M_j\left(\varphi \right)\left(H\right)={max}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$, $m_j\left(\varphi \right)\left(H\right)={min}_{T\in {𝒞}_j\left(H\right)}\varphi \left(T\right)$. It is proved that both $M_{p-\omega }\left(\varphi \right)$ and...

A graph $G$ is a $\{d,d+k\}$-graph, if one vertex has degree $d+k$ and the remaining vertices of $G$ have degree $d$. In the special case of $k=0$, the graph $G$ is $d$-regular. Let $k,p\ge 0$ and $d,n\ge 1$ be integers such that $n$ and $p$ are of the same parity. If $G$ is a connected $\{d,d+k\}$-graph of order $n$ without a matching $M$ of size $2\left|M\right|=n-p$, then we show in this paper the following: If $d=2$, then $k\ge 2(p+2)$ and (i) $n\ge k+p+6$. If $d\ge 3$ is odd and $t$ an integer with $1\le t\le p+2$, then (ii) $n\ge d+k+1$ for $k\ge d(p+2)$, (iii) $n\ge d(p+3)+2t+1$ for $d(p+2-t)+t\le k\le d(p+3-t)+t-3$, (iv) $n\ge d(p+3)+2p+7$ for $k\le p$. If $d\ge 4$ is even, then (v) $n\ge d+k+2-\eta$ for $k\ge d(p+3)+p+4+\eta$, (vi) $n\ge d+k+p+2-2t=d(p+4)+p+6$ for $k=d(p+3)+4+2t$ and $p\ge 1$,...

A $(p,q)$-sigraph $S$ is an ordered pair $(G,s)$ where $G=(V,E)$ is a $(p,q)$-graph and $s$ is a function which assigns to each edge of $G$ a positive or a negative sign. Let the sets $E^{+}$ and $E^{-}$ consist of $m$ positive and $n$ negative edges of $G$, respectively, where $m+n=q$. Given positive integers $k$ and $d$, $S$ is said to be $(k,d)$-graceful if the vertices of $G$ can be labeled with distinct integers from the set $\{0,1,\cdots ,k+(q-1\left)d\right\}$ such that when each edge $uv$ of $G$ is assigned the product of its sign and the absolute difference of the integers assigned to...